Date: April 17th, 1999
This is a loose collection of examples of computations with permutation characters and possible permutation characters in the GAP system [GAP21]. We mainly use the GAP implementation of the algorithms to compute possible permutation characters that are described in [BP98], and information from the Atlas of Finite Groups [CCN+85]. A possible permutation character of a finite group \(G\) is a character satisfying the conditions listed in Section "Possible Permutation Characters" of the GAP Reference Manual.
Section 8.16-1 was added in June 2009.
Section 8.16-2 was added in September 2009.
Section 8.16-3 was added in October 2009.
Section 8.16-4 was added in November 2009.
Section 8.17 was added in June 2012.
Section 8.18 was added in October 2017.
Section 8.19 was added in December 2021.
In the following, the GAP Character Table Library [Bre22] will be used frequently.
gap> LoadPackage( "ctbllib", "1.2", false ); true
We start with the sporadic simple Mathieu group \(G = M_{24}\) in its natural action on \(24\) points.
gap> g:= MathieuGroup( 24 );; gap> SetName( g, "m24" ); gap> Size( g ); IsSimple( g ); NrMovedPoints( g ); 244823040 true 24
The conjugacy classes that are computed for a group can be ordered differently in different GAP sessions. In order to make the output shown in the following examples stable, we first sort the conjugacy classes of \(G\) for our purposes.
gap> ccl:= AttributeValueNotSet( ConjugacyClasses, g );; gap> HasConjugacyClasses( g ); false gap> invariants:= List( ccl, c -> [ Order( Representative( c ) ), > Size( c ), Size( ConjugacyClass( g, Representative( c )^2 ) ) ] );; gap> SortParallel( invariants, ccl ); gap> SetConjugacyClasses( g, ccl );
The permutation character pi of \(G\) corresponding to the action on the moved points is constructed. This action is \(5\)-transitive.
gap> NrConjugacyClasses( g ); 26 gap> pi:= NaturalCharacter( g ); Character( CharacterTable( m24 ), [ 24, 8, 0, 6, 0, 0, 4, 0, 4, 2, 0, 3, 3, 2, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1 ] ) gap> IsTransitive( pi ); Transitivity( pi ); true 5 gap> Display( pi ); CT1 2 10 10 9 3 3 7 7 5 2 3 3 1 1 4 2 . 2 2 1 3 3 1 1 3 2 1 . 1 1 1 1 1 1 . . . 1 1 . 5 1 . 1 1 . . . . 1 . . . . . 1 . . . . 7 1 1 . . 1 . . . . . . 1 1 . . . . . 1 11 1 . . . . . . . . . . . . . . 1 . . . 23 1 . . . . . . . . . . . . . . . . . . 1a 2a 2b 3a 3b 4a 4b 4c 5a 6a 6b 7a 7b 8a 10a 11a 12a 12b 14a Y.1 24 8 . 6 . . 4 . 4 2 . 3 3 2 . 2 . . 1 2 1 . . . . . . 3 . 1 1 1 1 . . 5 . 1 1 . . . . 7 1 . . 1 1 . . 11 . . . . . . . 23 . . . . . 1 1 14b 15a 15b 21a 21b 23a 23b Y.1 1 1 1 . . 1 1
pi determines the permutation characters of the \(G\)-actions on related sets, for example piop on the set of ordered and piup on the set of unordered pairs of points.
gap> piop:= pi * pi; Character( CharacterTable( m24 ), [ 576, 64, 0, 36, 0, 0, 16, 0, 16, 4, 0, 9, 9, 4, 0, 4, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1 ] ) gap> IsTransitive( piop ); false gap> piup:= SymmetricParts( UnderlyingCharacterTable(pi), [ pi ], 2 )[1]; Character( CharacterTable( m24 ), [ 300, 44, 12, 21, 0, 4, 12, 0, 10, 5, 0, 6, 6, 4, 2, 3, 1, 0, 2, 2, 1, 1, 0, 0, 1, 1 ] ) gap> IsTransitive( piup ); false
Clearly the action on unordered pairs is not transitive, since the pairs \([ i, i ]\) form an orbit of their own. There are exactly two \(G\)-orbits on the unordered pairs, hence the \(G\)-action on \(2\)-sets of points is transitive.
gap> ScalarProduct( piup, TrivialCharacter( g ) ); 2 gap> comb:= Combinations( [ 1 .. 24 ], 2 );; gap> hom:= ActionHomomorphism( g, comb, OnSets );; gap> pihom:= NaturalCharacter( hom ); Character( CharacterTable( m24 ), [ 276, 36, 12, 15, 0, 4, 8, 0, 6, 3, 0, 3, 3, 2, 2, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0 ] ) gap> Transitivity( pihom ); 1
In terms of characters, the permutation character pihom is the difference of piup and pi . Note that GAP does not know that this difference is in fact a character; in general this question is not easy to decide without knowing the irreducible characters of \(G\), and up to now GAP has not computed the irreducibles.
gap> pi2s:= piup - pi; VirtualCharacter( CharacterTable( m24 ), [ 276, 36, 12, 15, 0, 4, 8, 0, 6, 3, 0, 3, 3, 2, 2, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0 ] ) gap> pi2s = pihom; true gap> HasIrr( g ); HasIrr( CharacterTable( g ) ); false false
The point stabilizer in the action on \(2\)-sets is in fact a maximal subgroup of \(G\), which is isomorphic to the automorphism group \(M_{22}:2\) of the Mathieu group \(M_{22}\). Thus this permutation action is primitive. But we cannot apply IsPrimitive (Reference: IsPrimitive) to the character pihom for getting this answer because primitivity of characters is defined in a different way, cf. IsPrimitiveCharacter (Reference: IsPrimitiveCharacter).
gap> IsPrimitive( g, comb, OnSets ); true
We could also have computed the transitive permutation character of degree \(276\) using the GAP Character Table Library instead of the group \(G\), since the character tables of \(G\) and all its maximal subgroups are available, together with the class fusions of the maximal subgroups into \(G\).
gap> tbl:= CharacterTable( "M24" ); CharacterTable( "M24" ) gap> maxes:= Maxes( tbl ); [ "M23", "M22.2", "2^4:a8", "M12.2", "2^6:3.s6", "L3(4).3.2_2", "2^6:(psl(3,2)xs3)", "L2(23)", "L3(2)" ] gap> s:= CharacterTable( maxes[2] ); CharacterTable( "M22.2" ) gap> TrivialCharacter( s )^tbl; Character( CharacterTable( "M24" ), [ 276, 36, 12, 15, 0, 4, 8, 0, 6, 3, 0, 3, 3, 2, 2, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0 ] )
Note that the sequence of conjugacy classes in the library table of \(G\) does in general not agree with the succession computed for the group.
We compute all possible permutation characters of the Mathieu group \(M_{11}\), using the three different strategies available in GAP. First we try the algorithm that enumerates all candidates via solving a system of inequalities, which is described in [BP98, Section 3.2].
gap> m11:= CharacterTable( "M11" );; gap> SetName( m11, "m11" ); gap> perms:= PermChars( m11 ); [ Character( m11, [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( m11, [ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ), Character( m11, [ 12, 4, 3, 0, 2, 1, 0, 0, 1, 1 ] ), Character( m11, [ 22, 6, 4, 2, 2, 0, 0, 0, 0, 0 ] ), Character( m11, [ 55, 7, 1, 3, 0, 1, 1, 1, 0, 0 ] ), Character( m11, [ 66, 10, 3, 2, 1, 1, 0, 0, 0, 0 ] ), Character( m11, [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ), Character( m11, [ 110, 6, 2, 6, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 110, 14, 2, 2, 0, 2, 0, 0, 0, 0 ] ), Character( m11, [ 132, 12, 6, 0, 2, 0, 0, 0, 0, 0 ] ), Character( m11, [ 144, 0, 0, 0, 4, 0, 0, 0, 1, 1 ] ), Character( m11, [ 165, 13, 3, 1, 0, 1, 1, 1, 0, 0 ] ), Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ), Character( m11, [ 220, 12, 4, 4, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ), Character( m11, [ 330, 2, 6, 2, 0, 2, 0, 0, 0, 0 ] ), Character( m11, [ 330, 18, 6, 2, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 396, 12, 0, 4, 1, 0, 0, 0, 0, 0 ] ), Character( m11, [ 440, 8, 8, 0, 0, 2, 0, 0, 0, 0 ] ), Character( m11, [ 440, 24, 8, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 495, 15, 0, 3, 0, 0, 1, 1, 0, 0 ] ), Character( m11, [ 660, 4, 3, 4, 0, 1, 0, 0, 0, 0 ] ), Character( m11, [ 660, 12, 3, 0, 0, 3, 0, 0, 0, 0 ] ), Character( m11, [ 660, 12, 12, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 660, 28, 3, 0, 0, 1, 0, 0, 0, 0 ] ), Character( m11, [ 720, 0, 0, 0, 0, 0, 0, 0, 5, 5 ] ), Character( m11, [ 792, 24, 0, 0, 2, 0, 0, 0, 0, 0 ] ), Character( m11, [ 880, 0, 16, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 990, 6, 0, 2, 0, 0, 2, 2, 0, 0 ] ), Character( m11, [ 990, 6, 0, 6, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 990, 30, 0, 2, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 1320, 8, 6, 0, 0, 2, 0, 0, 0, 0 ] ), Character( m11, [ 1320, 24, 6, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 1584, 0, 0, 0, 4, 0, 0, 0, 0, 0 ] ), Character( m11, [ 1980, 12, 0, 4, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 1980, 36, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 2640, 0, 12, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 3960, 24, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11, [ 7920, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> Length( perms ); 39
Next we try the improved combinatorial approach that is sketched at the end of Section 3.2 in [BP98]. We get the same characters, except that they may be ordered in a different way; thus we compare the ordered lists.
gap> degrees:= DivisorsInt( Size( m11 ) );; gap> perms2:= [];; gap> for d in degrees do > Append( perms2, PermChars( m11, d ) ); > od; gap> Set( perms ) = Set( perms2 ); true
Finally, we try the algorithm that is based on Gaussian elimination and that is described in [BP98, Section 3.3].
gap> perms3:= [];; gap> for d in degrees do > Append( perms3, PermChars( m11, rec( torso:= [ d ] ) ) ); > od; gap> Set( perms ) = Set( perms3 ); true
GAP provides two more functions to test properties of permutation characters. The first one yields no new information in our case, but the second excludes one possible permutation character; note that TestPerm5 needs a \(p\)-modular Brauer table, and the GAP character table library contains all Brauer tables of \(M_{11}\).
gap> newperms:= TestPerm4( m11, perms );; gap> newperms = perms; true gap> newperms:= TestPerm5( m11, perms, m11 mod 11 );; gap> newperms = perms; false gap> Difference( perms, newperms ); [ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ) ]
GAP knows the table of marks of \(M_{11}\), from which the permutation characters can be extracted. It turns out that \(M_{11}\) has \(39\) conjugacy classes of subgroups but only \(36\) different permutation characters, so three candidates computed above are in fact not permutation characters.
gap> tom:= TableOfMarks( "M11" ); TableOfMarks( "M11" ) gap> trueperms:= PermCharsTom( m11, tom );; gap> Length( trueperms ); Length( Set( trueperms ) ); 39 36 gap> Difference( perms, trueperms ); [ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ), Character( m11, [ 660, 4, 3, 4, 0, 1, 0, 0, 0, 0 ] ), Character( m11, [ 660, 12, 3, 0, 0, 3, 0, 0, 0, 0 ] ) ]
We are interested in the permutation character of \(U_6(2)\) (see [CCN+85, p. 115]) that corresponds to the action on the cosets of a \(M_{22}\) subgroup (see [CCN+85, p. 39]). The character tables of both the group and the point stabilizer are available in the GAP character table library, so we can compute class fusion and permutation character directly; note that if the class fusion is not stored on the table of the subgroup, in general one will not get a unique fusion but only a list of candidates for the fusion.
gap> u62:= CharacterTable( "U6(2)" );; gap> m22:= CharacterTable( "M22" );; gap> fus:= PossibleClassFusions( m22, u62 ); [ [ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 33, 34 ], [ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 34, 33 ], [ 1, 3, 7, 11, 14, 15, 22, 24, 24, 27, 33, 34 ], [ 1, 3, 7, 11, 14, 15, 22, 24, 24, 27, 34, 33 ], [ 1, 3, 7, 12, 14, 15, 22, 24, 24, 28, 33, 34 ], [ 1, 3, 7, 12, 14, 15, 22, 24, 24, 28, 34, 33 ] ] gap> RepresentativesFusions( m22, fus, u62 ); [ [ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 33, 34 ] ]
We see that there are six possible class fusions that are equivalent under table automorphisms of \(U_6(2)\) and \(M22\).
gap> cand:= Set( fus, > x -> Induced( m22, u62, [ TrivialCharacter( m22 ) ], x )[1] ); [ Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 0, 48, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( u62, cand ).ATLAS; [ "1a+22a+252a+616a+1155c+1386a+8064a+9240c", "1a+22a+252a+616a+1155b+1386a+8064a+9240b", "1a+22a+252a+616a+1155a+1386a+8064a+9240a" ] gap> aut:= AutomorphismsOfTable( u62 );; Size( aut ); 24 gap> elms:= Filtered( Elements( aut ), x -> Order( x ) = 3 ); [ (10,11,12)(26,27,28)(40,41,42), (10,12,11)(26,28,27)(40,42,41) ] gap> Position( cand, Permuted( cand[1], elms[1] ) ); 3 gap> Position( cand, Permuted( cand[3], elms[1] ) ); 2
The six fusions induce three different characters, they are conjugate under the action of the unique subgroup of order \(3\) in the group of table automorphisms of \(U_6(2)\). The table automorphisms of order \(3\) are induced by group automorphisms of \(U_6(2)\) (see [CCN+85, p. 120]). As can be seen from the list of maximal subgroups of \(U_6(2)\) in [CCN+85, p. 115], the three induced characters are in fact permutation characters which belong to the three classes of maximal subgroups of type \(M_{22}\) in \(U_6(2)\), which are permuted by an outer automorphism of order 3. Now we want to compute the extension of the above permutation character to the group \(U_6(2).2\), which corresponds to the action of this group on the cosets of a \(M_{22}.2\) subgroup.
gap> u622:= CharacterTable( "U6(2).2" );; gap> m222:= CharacterTable( "M22.2" );; gap> fus:= PossibleClassFusions( m222, u622 ); [ [ 1, 3, 7, 10, 13, 14, 20, 22, 22, 24, 29, 38, 39, 42, 41, 46, 50, 53, 58, 59, 59 ] ] gap> cand:= Induced( m222, u622, [ TrivialCharacter( m222 ) ], fus[1] ); [ Character( CharacterTable( "U6(2).2" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1080, 72, 0, 48, 8, 0, 0, 0, 18, 0, 0, 0, 8, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( u622, cand ).ATLAS; [ "1a+22a+252a+616a+1155a+1386a+8064a+9240a" ]
We see that for the embedding of \(M_{22}.2\) into \(U_6(2).2\), the class fusion is unique, so we get a unique extension of one of the above permutation characters. This implies that exactly one class of maximal subgroups of type \(M_{22}\) extends to \(M_{22}.2\) in a given group \(U_6(2).2\).
Now we show an alternative way to compute the characters dealt with in the previous example. This works also if the character table of the point stabilizer is not available. In this situation we can compute all those characters that have certain properties of permutation characters. Of course this may take much longer than the above computations, which needed only a few seconds. (The following calculations may need several hours, depending on the computer used.)
gap> cand:= PermChars( u62, rec( torso := [ 20736 ] ) ); [ Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 0, 48, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 0, 16, 6, 0, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
For the next step, that is, the computation of the extension of the permutation character to \(U_6(2).2\), we may use the above information, since the values on the inner classes are prescribed. The question which of the three candidates for \(U_6(2)\) extends to \(U_6(2).2\) depends on the choice of the class fusion of \(U_6(2)\) into \(U_6(2).2\). With respect to the class fusion that is stored on the GAP library table, the third candidate extends, as can be seen from the fact that this one is invariant under the permutation of conjugacy classes of \(U_6(2)\) that is induced by the action of the chosen supergroup \(U_6(2).2\).
gap> u622:= CharacterTable( "U6(2).2" );; gap> inv:= InverseMap( GetFusionMap( u62, u622 ) ); [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, [ 11, 12 ], 13, 14, 15, [ 16, 17 ], 18, 19, 20, 21, 22, 23, 24, 25, 26, [ 27, 28 ], [ 29, 30 ], 31, 32, [ 33, 34 ], [ 35, 36 ], 37, [ 38, 39 ], 40, [ 41, 42 ], 43, 44, [ 45, 46 ] ] gap> ext:= List( cand, x -> CompositionMaps( x, inv ) ); [ [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, [ 0, 48 ], 0, 16, 6, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, [ 0, 4 ], 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, [ 0, 48 ], 0, 16, 6, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, [ 0, 4 ], 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ] gap> cand:= PermChars( u622, rec( torso:= ext[3] ) ); [ Character( CharacterTable( "U6(2).2" ), [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1080, 72, 0, 48, 8, 0, 0, 0, 18, 0, 0, 0, 8, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0 ] ) ]
The group \(O_8^+(3)\) (see [CCN+85, p. 140]) contains a subgroup of type \(2^{{3+6}}.L_3(2)\), which extends to a maximal subgroup \(U\) in \(O_8^+(3).3\). For the computation of the permutation character, we cannot use explicit induction since the table of \(U\) is not available in the GAP table library. Since \(U \cap O_8^+(3)\) is contained in a \(O_8^+(2)\) subgroup of \(O_8^+(3)\), we can try to find the permutation character of \(O_8^+(2)\) corresponding to the action on the cosets of \(U \cap O_8^+(3)\), and then induce this character to \(O_8^+(3)\). This kind of computations becomes more difficult with increasing degree, so we try to reduce the problem further. In fact, the \(2^{{3+6}}.L_3(2)\) group is contained in a \(2^6:A_8\) subgroup of \(O_8^+(2)\), in which the index is only \(15\); the unique possible permutation character of this degree can be read off immediately. Induction to \(O_8^+(3)\) through the chain of subgroups is possible provided the class fusions are available. There are \(24\) possible fusions from \(O_8^+(2)\) into \(O_8^+(3)\), which are all equivalent w.r.t. table automorphisms of \(O_8^+(3)\). If we later want to consider the extension of the permutation character in question to \(O_8^+(3).3\) then we have to choose a fusion of an \(O_8^+(2)\) subgroup that does not extend to \(O_8^+(2).3\). But if for example our question is just whether the resulting permutation character is multiplicity-free then this can be decided already from the permutation character of \(O_8^+(3)\).
gap> o8p3:= CharacterTable("O8+(3)");; gap> Size( o8p3 ) / (2^9*168); 57572775 gap> o8p2:= CharacterTable( "O8+(2)" );; gap> fus:= PossibleClassFusions( o8p2, o8p3 );; gap> Length( fus ); 24 gap> rep:= RepresentativesFusions( o8p2, fus, o8p3 ); [ [ 1, 5, 2, 3, 4, 5, 7, 8, 12, 16, 17, 19, 23, 20, 21, 22, 23, 24, 25, 26, 37, 38, 42, 31, 32, 36, 49, 52, 51, 50, 43, 44, 45, 53, 55, 56, 57, 71, 71, 71, 72, 73, 74, 78, 79, 83, 88, 89, 90, 94, 100, 101, 105 ] ] gap> fus:= rep[1];; gap> Size( o8p2 ) / (2^9*168); 2025 gap> sub:= CharacterTable( "2^6:A8" );; gap> subfus:= GetFusionMap( sub, o8p2 ); [ 1, 3, 2, 2, 4, 5, 6, 13, 3, 6, 12, 13, 14, 7, 21, 24, 11, 30, 29, 31, 13, 17, 15, 16, 14, 17, 36, 37, 18, 41, 24, 44, 48, 28, 33, 32, 34, 35, 35, 51, 51 ] gap> fus:= CompositionMaps( fus, subfus ); [ 1, 2, 5, 5, 3, 4, 5, 23, 2, 5, 19, 23, 20, 7, 37, 31, 17, 50, 51, 43, 23, 23, 21, 22, 20, 23, 56, 57, 24, 72, 31, 78, 89, 52, 45, 44, 53, 55, 55, 100, 100 ] gap> Size( sub ) / (2^9*168); 15 gap> List( Irr( sub ), Degree ); [ 1, 7, 14, 20, 21, 21, 21, 28, 35, 45, 45, 56, 64, 70, 28, 28, 35, 35, 35, 35, 70, 70, 70, 70, 140, 140, 140, 140, 140, 210, 210, 252, 252, 280, 280, 315, 315, 315, 315, 420, 448 ] gap> cand:= PermChars( sub, 15 ); [ Character( CharacterTable( "2^6:A8" ), [ 15, 15, 15, 7, 7, 7, 7, 7, 3, 3, 3, 3, 3, 0, 0, 0, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0 ] ) ] gap> ind:= Induced( sub, o8p3, cand, fus ); [ Character( CharacterTable( "O8+(3)" ), [ 57572775, 59535, 59535, 59535, 3591, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2187, 0, 27, 135, 135, 135, 243, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 27, 27, 0, 0, 0, 0, 27, 27, 27, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> o8p33:= CharacterTable( "O8+(3).3" );; gap> inv:= InverseMap( GetFusionMap( o8p3, o8p33 ) ); [ 1, [ 2, 3, 4 ], 5, 6, [ 7, 8, 9 ], [ 10, 11, 12 ], 13, [ 14, 15, 16 ], 17, 18, 19, [ 20, 21, 22 ], 23, [ 24, 25, 26 ], [ 27, 28, 29 ], 30, [ 31, 32, 33 ], [ 34, 35, 36 ], [ 37, 38, 39 ], [ 40, 41, 42 ], [ 43, 44, 45 ], 46, [ 47, 48, 49 ], 50, [ 51, 52, 53 ], 54, 55, 56, 57, [ 58, 59, 60 ], [ 61, 62, 63 ], 64, [ 65, 66, 67 ], 68, [ 69, 70, 71 ], [ 72, 73, 74 ], [ 75, 76, 77 ], [ 78, 79, 80 ], [ 81, 82, 83 ], 84, 85, [ 86, 87, 88 ], [ 89, 90, 91 ], [ 92, 93, 94 ], 95, 96, [ 97, 98, 99 ], [ 100, 101, 102 ], [ 103, 104, 105 ], [ 106, 107, 108 ], [ 109, 110, 111 ], [ 112, 113, 114 ] ] gap> ext:= CompositionMaps( ind[1], inv ); [ 57572775, 59535, 3591, 0, 0, 0, 0, 0, 2187, 0, 27, 135, 243, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 27, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] gap> perms:= PermChars( o8p33, rec( torso:= ext ) ); [ Character( CharacterTable( "O8+(3).3" ), [ 57572775, 59535, 3591, 0, 0, 0, 0, 0, 2187, 0, 27, 135, 243, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 27, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3159, 3159, 243, 243, 39, 39, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( o8p33, perms ).ATLAS; [ "1a+780aabb+2457a+2808abc+9450aaabbcc+18200abcdddef+24192a+54600a^{5\ }b+70200aabb+87360ab+139776a^{5}+147420a^{4}b^{4}+163800ab+184275aabc+\ 199017aa+218700a+245700a+291200aef+332800a^{4}b^{5}c^{5}+491400aaabcd+\ 531441a^{5}b^{4}c^{4}+552825a^{4}+568620aabb+698880a^{4}b^{4}+716800aa\ abbccdddeeff+786240aabb+873600aa+998400aa+1257984a^{6}+1397760aa" ]
We want to know whether the permutation character of \(O_7(3).2\) (see [CCN+85, p. 108]) on the cosets of its maximal subgroup \(U\) of type \(2^7.S_7\) is multiplicity-free. As in the previous examples, first we try to compute the permutation character of the simple group \(O_7(3)\). It turns out that the direct computation of all candidates from the degree is very time consuming. But we can use for example the additional information provided by the fact that \(U\) contains an \(A_7\) subgroup. We compute the possible class fusions.
gap> o73:= CharacterTable( "O7(3)" );; gap> a7:= CharacterTable( "A7" );; gap> fus:= PossibleClassFusions( a7, o73 ); [ [ 1, 3, 6, 10, 15, 16, 24, 33, 33 ], [ 1, 3, 7, 10, 15, 16, 22, 33, 33 ] ]
We cannot decide easily which fusion is the right one, but already the fact that no other fusions are possible gives us some information about impossible constituents of the permutation character we want to compute.
gap> ind:= List( fus, > x -> Induced( a7, o73, [ TrivialCharacter( a7 ) ], x )[1] );; gap> mat:= MatScalarProducts( o73, Irr( o73 ), ind );; gap> sum:= Sum( mat ); [ 2, 6, 2, 0, 8, 6, 2, 4, 4, 8, 3, 0, 4, 4, 9, 3, 5, 0, 0, 9, 0, 10, 5, 6, 15, 1, 12, 1, 15, 7, 2, 4, 14, 16, 0, 12, 12, 7, 8, 8, 14, 12, 12, 14, 6, 6, 20, 16, 12, 12, 12, 10, 10, 12, 12, 8, 12, 6 ] gap> const:= Filtered( [ 1 .. Length( sum ) ], x -> sum[x] <> 0 ); [ 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58 ] gap> Length( const ); 52 gap> const:= Irr( o73 ){ const };; gap> rat:= RationalizedMat( const );;
But much more can be deduced from the fact that certain zeros of the permutation character can be predicted.
gap> names:= ClassNames( o73 ); [ "1a", "2a", "2b", "2c", "3a", "3b", "3c", "3d", "3e", "3f", "3g", "4a", "4b", "4c", "4d", "5a", "6a", "6b", "6c", "6d", "6e", "6f", "6g", "6h", "6i", "6j", "6k", "6l", "6m", "6n", "6o", "6p", "7a", "8a", "8b", "9a", "9b", "9c", "9d", "10a", "10b", "12a", "12b", "12c", "12d", "12e", "12f", "12g", "12h", "13a", "13b", "14a", "15a", "18a", "18b", "18c", "18d", "20a" ] gap> List( fus, x -> names{ x } ); [ [ "1a", "2b", "3b", "3f", "4d", "5a", "6h", "7a", "7a" ], [ "1a", "2b", "3c", "3f", "4d", "5a", "6f", "7a", "7a" ] ] gap> torso:= [ 28431 ];; gap> zeros:= [ 5, 8, 9, 11, 17, 20, 23, 28, 29, 32, 36, 37, 38, > 43, 46, 47, 48, 53, 54, 55, 56, 57, 58 ];; gap> names{ zeros }; [ "3a", "3d", "3e", "3g", "6a", "6d", "6g", "6l", "6m", "6p", "9a", "9b", "9c", "12b", "12e", "12f", "12g", "15a", "18a", "18b", "18c", "18d", "20a" ]
Every order \(3\) element of \(U\) lies in an \(A_7\) subgroup of \(U\), so among the classes of element order \(3\), at most the classes 3B, 3C, and 3F can have nonzero permutation character values. The excluded classes of element order \(6\) are the square roots of the excluded order \(3\) elements, likewise the given classes of element orders \(9\), \(12\), and \(18\) are excluded. The character value on 20A must be zero because \(U\) does not contain elements of this order. So we enter the additional information about these zeros.
gap> for i in zeros do > torso[i]:= 0; > od; gap> torso; [ 28431,,,, 0,,, 0, 0,, 0,,,,,, 0,,, 0,,, 0,,,,, 0, 0,,, 0,,,, 0, 0, 0,,,,, 0,,, 0, 0, 0,,,,, 0, 0, 0, 0, 0, 0 ] gap> perms:= PermChars( o73, rec( torso:= torso, chars:= rat ) ); [ Character( CharacterTable( "O7(3)" ), [ 28431, 567, 567, 111, 0, 0, 243, 0, 0, 81, 0, 15, 3, 27, 15, 6, 0, 0, 27, 0, 3, 27, 0, 0, 0, 3, 9, 0, 0, 3, 3, 0, 4, 1, 1, 0, 0, 0, 0, 2, 2, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( o73, perms ).ATLAS; [ "1a+78a+168a+182a+260ab+1092a+2457a+2730a+4095b+5460a+11648a" ]
We see that this character is already multiplicity free, so this holds also for its extension to \(O_7(3).2\), and we need not compute this extension. (Of course we could compute it in the same way as in the examples above.)
We are interested in the permutation character of \(O_8^+(3).2_1\) that corresponds to the action on the cosets of a subgroup of type \(2^7.A_8\). The intersection of the point stabilizer with the simple group \(O_8^+(3)\) is of type \(2^6.A_8\). First we compute the class fusion of these groups, modulo problems with ambiguities due to table automorphisms.
gap> o8p3:= CharacterTable( "O8+(3)" );; gap> o8p2:= CharacterTable( "O8+(2)" );; gap> fus:= PossibleClassFusions( o8p2, o8p3 );; gap> NamesOfFusionSources( o8p2 ); [ "A9", "2^8:O8+(2)", "(D10xD10).2^2", "(3x3^3:S3):2", "(3x3^(1+2)+:2A4).2", "2^(3+3+3).L3(2)", "NRS(O8+(2),2^(3+3+3)_a)", "NRS(O8+(2),2^(3+3+3)_b)", "O8+(2)N2", "O8+(2)M2", "O8+(2)M3", "O8+(2)M5", "O8+(2)M6", "O8+(2)M8", "O8+(2)M9", "(3xU4(2)):2", "O8+(2)M11", "O8+(2)M12", "2^(1+8)_+:(S3xS3xS3)", "3^4:2^3.S4(a)", "(A5xA5):2^2", "O8+(2)M16", "O8+(2)M17", "2^(1+8)+.O8+(2)", "7:6", "(A5xD10).2", "(D10xA5).2", "O8+(2)N5C", "2^6:A8", "2.O8+(2)", "2^2.O8+(2)", "S6(2)" ] gap> sub:= CharacterTable( "2^6:A8" );; gap> subfus:= GetFusionMap( sub, o8p2 ); [ 1, 3, 2, 2, 4, 5, 6, 13, 3, 6, 12, 13, 14, 7, 21, 24, 11, 30, 29, 31, 13, 17, 15, 16, 14, 17, 36, 37, 18, 41, 24, 44, 48, 28, 33, 32, 34, 35, 35, 51, 51 ] gap> fus:= List( fus, x -> CompositionMaps( x, subfus ) );; gap> fus:= Set( fus );; gap> Length( fus ); 24
The ambiguities due to Galois automorphisms disappear when we are looking for the permutation characters induced by the fusions.
gap> ind:= List( fus, x -> Induced( sub, o8p3, > [ TrivialCharacter( sub ) ], x )[1] );; gap> ind:= Set( ind );; gap> Length( ind ); 6
Now we try to extend the candidates to \(O_8^+(3).2_1\); the choice of the fusion of \(O_8^+(3)\) into \(O_8^+(3).2_1\) determines which of the candidates may extend.
gap> o8p32:= CharacterTable( "O8+(3).2_1" );; gap> fus:= GetFusionMap( o8p3, o8p32 );; gap> ext:= List( ind, x -> CompositionMaps( x, InverseMap( fus ) ) );; gap> ext:= Filtered( ext, x -> ForAll( x, IsInt ) ); [ [ 3838185, 17577, 8505, 8505, 873, 0, 0, 0, 0, 6561, 0, 0, 729, 0, 9, 105, 45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 9, 27, 27, 0, 0, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0 ], [ 3838185, 17577, 8505, 8505, 873, 0, 6561, 0, 0, 0, 0, 0, 729, 0, 9, 105, 45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 0, 0, 0, 9, 27, 27, 0, 0, 9, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ]
We compute the extensions of the first candidate; the other belongs to another class of subgroups, which is the image under an outer automorphism.
gap> perms:= PermChars( o8p32, rec( torso:= ext[1] ) ); [ Character( CharacterTable( "O8+(3).2_1" ), [ 3838185, 17577, 8505, 8505, 873, 0, 0, 0, 0, 6561, 0, 0, 729, 0, 9, 105, 45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 9, 27, 27, 0, 0, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 3159, 1575, 567, 63, 87, 15, 0, 0, 45, 0, 81, 9, 27, 0, 0, 3, 3, 3, 3, 5, 5, 0, 0, 0, 4, 0, 0, 27, 0, 9, 0, 0, 15, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( o8p32, perms ).ATLAS; [ "1a+260abc+520ab+819a+2808b+9450aab+18200a+23400ac+29120b+36400aab+4\ 6592abce+49140d+66339a+98280ab+163800a+189540d+232960d+332800ab+368550\ a+419328a+531441ab" ]
Now we repeat the calculations for \(O_8^+(3).2_2\) instead of \(O_8^+(3).2_1\).
gap> o8p32:= CharacterTable( "O8+(3).2_2" );; gap> fus:= GetFusionMap( o8p3, o8p32 );; gap> ext:= List( ind, x -> CompositionMaps( x, InverseMap( fus ) ) );; gap> ext:= Filtered( ext, x -> ForAll( x, IsInt ) );; gap> perms:= PermChars( o8p32, rec( torso:= ext[1] ) ); [ Character( CharacterTable( "O8+(3).2_2" ), [ 3838185, 17577, 8505, 873, 0, 0, 0, 6561, 0, 0, 0, 0, 729, 0, 9, 105, 45, 105, 30, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 0, 9, 27, 0, 0, 0, 27, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 199017, 2025, 297, 441, 73, 9, 0, 1215, 0, 0, 0, 0, 0, 81, 0, 0, 0, 0, 27, 27, 0, 1, 9, 12, 0, 0, 45, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( o8p32, perms ).ATLAS; [ "1a+260aac+520ab+819a+2808a+9450aaa+18200accee+23400ac+29120a+36400a\ +46592aa+49140c+66339a+93184a+98280ab+163800a+184275ac+189540c+232960c\ +332800aa+419328a+531441aa" ]
We might be interested in the extension to \(O_8^+(3).(2^2)_{122}\). It is clear that this cannot be multiplicity free because of the multiplicity 9450aaa in the character induced from \(O_8^+(3).2_2\). We could put the extensions to the index two subgroups together, but it is simpler (and not expensive) to run the same program as above.
gap> o8p322:= CharacterTable( "O8+(3).(2^2)_{122}" );; gap> fus:= GetFusionMap( o8p32, o8p322 );; gap> ext:= List( perms, x -> CompositionMaps( x, InverseMap( fus ) ) );; gap> ext:= Filtered( ext, x -> ForAll( x, IsInt ) );; gap> perms:= PermChars( o8p322, rec( torso:= ext[1] ) ); [ Character( CharacterTable( "O8+(3).(2^2)_{122}" ), [ 3838185, 17577, 8505, 873, 0, 0, 0, 6561, 0, 0, 729, 0, 9, 105, 45, 105, 30, 0, 0, 0, 0, 0, 0, 189, 0, 0, 9, 9, 27, 0, 0, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 3159, 1575, 567, 63, 87, 15, 0, 0, 45, 0, 81, 9, 27, 0, 0, 3, 3, 3, 5, 0, 0, 4, 0, 0, 27, 0, 9, 0, 0, 15, 0, 3, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 199017, 2025, 297, 441, 73, 9, 0, 1215, 0, 0, 0, 0, 81, 0, 0, 0, 27, 27, 0, 1, 9, 12, 0, 0, 45, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 28431, 1647, 135, 63, 87, 39, 0, 0, 243, 27, 0, 0, 81, 63, 0, 0, 0, 9, 0, 3, 3, 6, 2, 0, 0, 0, 9, 0, 0, 3, 3, 3, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0 ] ) ] gap> PermCharInfo( o8p322, perms ).ATLAS; [ "1a+260ace+819a+1040a+2808c+9450aac+18200a+23400ae+29120c+36400aac+4\ 6592ac+49140g+66339a+93184a+163800b+189540g+196560a+232960g+332800ac+3\ 68550a+419328a+531441ac" ]
We want to know whether the permutation character corresponding to the action of \(S_4(4).4\) (see [CCN+85, p. 44]) on the cosets of its maximal subgroup of type \(5^2:[2^5]\) is multiplicity free. The library names of subgroups for which the class fusions are stored are listed as value of the attribute NamesOfFusionSources (Reference: NamesOfFusionSources), and for groups whose isomorphism type is not determined by the name this is the recommended way to find out whether the table of the subgroup is contained in the GAP library and known to belong to this group. (It might be that a table with such a name is contained in the library but belongs to another group, and it may also be that the table of the group is contained in the library --with any name-- but it is not known that this group is isomorphic to a subgroup of \(S_4(4).4\).)
gap> s444:= CharacterTable( "S4(4).4" );; gap> NamesOfFusionSources( s444 ); [ "(L3(2)xS4(4):2).2", "S4(4)", "S4(4).2" ]
So we cannot simply fetch the table of the subgroup. As in the previous examples, we compute the possible permutation characters.
gap> perms:= PermChars( s444, > rec( torso:= [ Size( s444 ) / ( 5^2*2^5 ) ] ) ); [ Character( CharacterTable( "S4(4).4" ), [ 4896, 384, 96, 0, 16, 32, 36, 16, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "S4(4).4" ), [ 4896, 192, 32, 0, 0, 8, 6, 1, 0, 2, 0, 0, 36, 0, 12, 0, 0, 0, 1, 0, 6, 6, 2, 2, 0, 0, 0, 0, 1, 1 ] ), Character( CharacterTable( "S4(4).4" ), [ 4896, 240, 64, 0, 8, 8, 36, 16, 0, 0, 0, 0, 0, 12, 8, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
So there are three candidates. None of them is multiplicity free, so we need not decide which of the candidates actually belongs to the group \(5^2:[2^5]\) we have in mind.
gap> PermCharInfo( s444, perms ).ATLAS; [ "1abcd+50abcd+153abcd+170a^{4}b^{4}+680aabb", "1a+50ac+153a+170aab+256a+680abb+816a+1020a", "1ac+50ac+68a+153abcd+170aabbb+204a+680abb+1020a" ]
(If we would be interested which candidate is the right one, we could for example look at the intersection with \(S_4(4)\), and hope for a contradiction to the fact that the group must lie in a \((A_5 \times A_5):2\) subgroup.)
We compute the permutation characters of the sporadic simple Conway group \(Co_1\) (see [CCN+85, p. 180]) corresponding to the actions on the cosets of involution centralizers. Equivalently, we are interested in the action of \(Co_1\) on conjugacy classes of involutions. These characters can be computed as follows. First we take the table of \(Co_1\).
gap> t:= CharacterTable( "Co1" ); CharacterTable( "Co1" )
The centralizer of each 2A element is a maximal subgroup of \(Co_1\). This group is also contained in the table library. So we can compute the permutation character by explicit induction, and the decomposition in irreducibles is computed with the command PermCharInfo (Reference: PermCharInfo).
gap> s:= CharacterTable( Maxes( t )[5] ); CharacterTable( "2^(1+8)+.O8+(2)" ) gap> ind:= Induced( s, t, [ TrivialCharacter( s ) ] );; gap> PermCharInfo( t, ind ).ATLAS; [ "1a+299a+17250a+27300a+80730a+313950a+644644a+2816856a+5494125a+1243\ 2420a+24794000a" ]
The centralizer of a 2B element is not maximal. First we compute which maximal subgroup can contain it. The character tables of all maximal subgroups of \(Co_1\) are contained in the GAP's table library, so we may take these tables and look at the group orders.
gap> centorder:= SizesCentralizers( t )[3];; gap> maxes:= List( Maxes( t ), CharacterTable );; gap> cand:= Filtered( maxes, x -> Size( x ) mod centorder = 0 ); [ CharacterTable( "(A4xG2(4)):2" ) ] gap> u:= cand[1];; gap> index:= Size( u ) / centorder; 3
So there is a unique class of maximal subgroups containing the centralizer of a 2B element, as a subgroup of index \(3\). We compute the unique permutation character of degree \(3\) of this group, and induce this character to \(G\).
gap> subperm:= PermChars( u, rec( degree := index, bounds := false ) ); [ Character( CharacterTable( "(A4xG2(4)):2" ), [ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ) ] gap> subperm = PermChars( u, rec( torso := [ 3 ] ) ); true gap> ind:= Induced( u, t, subperm ); [ Character( CharacterTable( "Co1" ), [ 2065694400, 181440, 119408, 38016, 2779920, 0, 0, 378, 30240, 864, 0, 720, 316, 80, 2520, 30, 0, 6480, 1508, 0, 0, 0, 0, 0, 38, 18, 105, 0, 600, 120, 56, 24, 0, 12, 0, 0, 0, 120, 48, 18, 0, 0, 6, 0, 360, 144, 108, 0, 0, 10, 0, 0, 0, 0, 0, 4, 2, 3, 9, 0, 0, 15, 3, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 12, 8, 0, 6, 0, 0, 3, 0, 1, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0 ] ) ] gap> PermCharInfo( t, ind ).ATLAS; [ "1a+1771a+8855a+27300aa+313950a+345345a+644644aa+871884aaa+1771000a+\ 2055625a+4100096a+7628985a+9669660a+12432420aa+21528000aa+23244375a+24\ 174150aa+24794000a+31574400aa+40370176a+60435375a+85250880aa+100725625\ a+106142400a+150732800a+184184000a+185912496a+207491625a+299710125a+30\ 2176875a" ]
Finally, we try the same for the centralizer of a 2C element.
gap> centorder:= SizesCentralizers( t )[4];; gap> cand:= Filtered( maxes, x -> Size( x ) mod centorder = 0 ); [ CharacterTable( "Co2" ), CharacterTable( "2^11:M24" ) ]
The group order excludes all except two classes of maximal subgroups. But the 2C centralizer cannot lie in \(Co_2\) because the involution centralizers in \(Co_2\) are too small.
gap> u:= cand[1];; gap> GetFusionMap( u, t ); [ 1, 2, 2, 4, 7, 6, 9, 11, 11, 10, 11, 12, 14, 17, 16, 21, 23, 20, 22, 22, 24, 28, 30, 33, 31, 32, 33, 33, 37, 42, 41, 43, 44, 48, 52, 49, 53, 55, 53, 52, 54, 60, 60, 60, 64, 65, 65, 67, 66, 70, 73, 72, 78, 79, 84, 85, 87, 92, 93, 93 ] gap> centorder; 389283840 gap> SizesCentralizers( u )[4]; 1474560
So we try the second candidate.
gap> u:= cand[2]; CharacterTable( "2^11:M24" ) gap> index:= Size( u ) / centorder; 1288 gap> subperm:= PermChars( u, rec( torso := [ index ] ) ); [ Character( CharacterTable( "2^11:M24" ), [ 1288, 1288, 1288, 56, 56, 56, 56, 56, 56, 48, 48, 48, 48, 48, 10, 10, 10, 10, 7, 7, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 0, 0, 0, 0, 2, 2, 2, 2, 3, 3, 3, 1, 1, 2, 2, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> subperm = PermChars( u, rec( degree:= index, bounds := false ) ); true gap> ind:= Induced( u, t, subperm ); [ Character( CharacterTable( "Co1" ), [ 10680579000, 1988280, 196560, 94744, 0, 17010, 0, 945, 7560, 3432, 2280, 1728, 252, 308, 0, 225, 0, 0, 0, 270, 0, 306, 0, 46, 45, 25, 0, 0, 120, 32, 12, 52, 36, 36, 0, 0, 0, 0, 0, 45, 15, 0, 9, 3, 0, 0, 0, 0, 18, 0, 30, 0, 6, 18, 0, 3, 5, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( t, ind ).ATLAS; [ "1a+17250aa+27300a+80730aa+644644aaa+871884a+1821600a+2055625aaa+281\ 6856a+5494125a^{4}+12432420aa+16347825aa+23244375a+24174150aa+24667500\ aa+24794000aaa+31574400a+40370176a+55255200a+66602250a^{4}+83720000aa+\ 85250880aaa+91547820aa+106142400a+150732800a+184184000aaa+185912496aaa\ +185955000aaa+207491625aaa+215547904aa+241741500aaa+247235625a+2578576\ 00aa+259008750a+280280000a+302176875a+326956500a+387317700a+402902500a\ +464257024a+469945476b+502078500a+503513010a+504627200a+522161640a" ]
We compute the multiplicity free possible permutation characters of \(G_2(3)\) (see [CCN+85, p. 60]). For each divisor \(d\) of the group order, we compute all those possible permutation characters of degree \(d\) of \(G\) for which each irreducible constituent occurs with multiplicity at most \(1\); this is done by prescribing the maxmult component of the second argument of PermChars (Reference: PermChars) to be the list with \(1\) at each position.
gap> t:= CharacterTable( "G2(3)" ); CharacterTable( "G2(3)" ) gap> t:= CharacterTable( "G2(3)" );; gap> n:= Length( RationalizedMat( Irr( t ) ) );; gap> maxmult:= List( [ 1 .. n ], i -> 1 );; gap> perms:= [];; gap> divs:= DivisorsInt( Size( t ) );; gap> for d in divs do > Append( perms, > PermChars( t, rec( bounds := false, > degree := d, > maxmult := maxmult ) ) ); > od; gap> Length( perms ); 42 gap> List( perms, Degree ); [ 1, 351, 351, 364, 364, 378, 378, 546, 546, 546, 546, 546, 702, 702, 728, 728, 1092, 1092, 1092, 1092, 1092, 1092, 1092, 1092, 1456, 1456, 1638, 1638, 2184, 2184, 2457, 2457, 2457, 2457, 3159, 3276, 3276, 3276, 3276, 4368, 6552, 6552 ]
For finding out which of these candidates are really permutation characters, we could inspect them piece by piece, using the information in [CCN+85]. For example, the candidates of degrees \(351\), \(364\), and \(378\) are induced from the trivial characters of maximal subgroups of \(G\), whereas the candidates of degree \(546\) are not permutation characters.
Since the table of marks of \(G\) is available in GAP, we can extract all permutation characters from the table of marks, and then filter out the multiplicity free ones.
gap> tom:= TableOfMarks( "G2(3)" ); TableOfMarks( "G2(3)" ) gap> tbl:= CharacterTable( "G2(3)" ); CharacterTable( "G2(3)" ) gap> permstom:= PermCharsTom( tbl, tom );; gap> Length( permstom ); 433 gap> multfree:= Intersection( perms, permstom );; gap> Length( multfree ); 15 gap> List( multfree, Degree ); [ 1, 351, 351, 364, 364, 378, 378, 702, 702, 728, 728, 1092, 1092, 2184, 2184 ]
We compute the primitive permutation characters of degree \(11\,200\) of \(O_8^+(2)\) and \(O_8^+(2).2\) (see [CCN+85, p. 85]). The character table of the maximal subgroup of type \(3^4:2^3.S_4\) in \(O_8^+(2)\) is not available in the GAP table library. But the group extends to a wreath product of \(S_3\) and \(S_4\) in the group \(O_8^+(2).2\), and the table of this wreath product can be constructed easily.
gap> tbl2:= CharacterTable("O8+(2).2");; gap> s3:= CharacterTable( "Symmetric", 3 );; gap> s:= CharacterTableWreathSymmetric( s3, 4 ); CharacterTable( "Sym(3)wrS4" )
The permutation character pi of \(O_8^+(2).2\) can thus be computed by explicit induction, and the character of \(O_8^+(2)\) is obtained by restriction of pi.
gap> fus:= PossibleClassFusions( s, tbl2 ); [ [ 1, 41, 6, 3, 48, 9, 42, 19, 51, 8, 5, 50, 24, 49, 7, 2, 44, 22, 42, 12, 53, 17, 58, 21, 5, 47, 26, 50, 37, 52, 23, 60, 18, 4, 46, 25, 14, 61, 20, 9, 53, 30, 51, 26, 64, 8, 52, 31, 13, 56, 38 ] ] gap> pi:= Induced( s, tbl2, [ TrivialCharacter( s ) ], fus[1] )[1]; Character( CharacterTable( "O8+(2).2" ), [ 11200, 256, 160, 160, 80, 40, 40, 76, 13, 0, 0, 8, 8, 4, 0, 0, 16, 16, 4, 4, 4, 1, 1, 1, 1, 5, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, 0, 1120, 96, 0, 16, 0, 16, 8, 10, 4, 6, 7, 12, 3, 0, 0, 2, 0, 4, 0, 1, 1, 0, 0, 1, 0, 0, 0 ] ) gap> PermCharInfo( tbl2, pi ).ATLAS; [ "1a+84a+168a+175a+300a+700c+972a+1400a+3200a+4200b" ] gap> tbl:= CharacterTable( "O8+(2)" ); CharacterTable( "O8+(2)" ) gap> rest:= RestrictedClassFunction( pi, tbl ); Character( CharacterTable( "O8+(2)" ), [ 11200, 256, 160, 160, 160, 80, 40, 40, 40, 76, 13, 0, 0, 8, 8, 8, 4, 0, 0, 0, 16, 16, 16, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 5, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0 ] ) gap> PermCharInfo( tbl, rest ).ATLAS; [ "1a+84abc+175a+300a+700bcd+972a+3200a+4200a" ]
We prove that the sporadic simple Mathieu group \(G = M_{22}\) (see [CCN+85, p. 39]) has no subgroup of index \(56\). In [Isa76], remark after Theorem 5.18, this is stated as an example of the case that a character may be a possible permutation character but not a permutation character. Let us consider the possible permutation character of degree \(56\) of \(G\).
gap> tbl:= CharacterTable( "M22" ); CharacterTable( "M22" ) gap> perms:= PermChars( tbl, rec( torso:= [ 56 ] ) ); [ Character( CharacterTable( "M22" ), [ 56, 8, 2, 4, 0, 1, 2, 0, 0, 2, 1, 1 ] ) ] gap> pi:= perms[1];; gap> Norm( pi ); 2 gap> Display( tbl, rec( chars:= perms ) ); M22 2 7 7 2 5 4 . 2 . . 3 . . 3 2 1 2 . . . 1 . . . . . 5 1 . . . . 1 . . . . . . 7 1 . . . . . . 1 1 . . . 11 1 . . . . . . . . . 1 1 1a 2a 3a 4a 4b 5a 6a 7a 7b 8a 11a 11b 2P 1a 1a 3a 2a 2a 5a 3a 7a 7b 4a 11b 11a 3P 1a 2a 1a 4a 4b 5a 2a 7b 7a 8a 11a 11b 5P 1a 2a 3a 4a 4b 1a 6a 7b 7a 8a 11a 11b 7P 1a 2a 3a 4a 4b 5a 6a 1a 1a 8a 11b 11a 11P 1a 2a 3a 4a 4b 5a 6a 7a 7b 8a 1a 1a Y.1 56 8 2 4 . 1 2 . . 2 1 1
Suppose that pi is a permutation character of \(G\). Since \(G\) is \(2\)-transitive on the \(56\) cosets of the point stabilizer \(S\), this stabilizer is transitive on \(55\) points, and thus \(G\) has a subgroup \(U\) of index \(56 \cdot 55 = 3080\). We compute the possible permutation character of this degree.
gap> perms:= PermChars( tbl, rec( torso:= [ 56 * 55 ] ) );; gap> Length( perms ); 16
\(U\) is contained in \(S\), so only those candidates must be considered that vanish on all classes where pi vanishes. Furthermore, the index of \(U\) in \(S\) is odd, so the Sylow \(2\) subgroups of \(U\) and \(S\) are isomorphic; \(S\) contains elements of order \(8\), hence also \(U\) does.
gap> OrdersClassRepresentatives( tbl ); [ 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 11, 11 ] gap> perms:= Filtered( perms, x -> x[5] = 0 and x[10] <> 0 ); [ Character( CharacterTable( "M22" ), [ 3080, 56, 2, 12, 0, 0, 2, 0, 0, 2, 0, 0 ] ), Character( CharacterTable( "M22" ), [ 3080, 8, 2, 8, 0, 0, 2, 0, 0, 4, 0, 0 ] ), Character( CharacterTable( "M22" ), [ 3080, 24, 11, 4, 0, 0, 3, 0, 0, 2, 0, 0 ] ), Character( CharacterTable( "M22" ), [ 3080, 24, 20, 4, 0, 0, 0, 0, 0, 2, 0, 0 ] ) ]
For getting an overview of the distribution of the elements of \(U\) to the conjugacy classes of \(G\), we use the output of PermCharInfo (Reference: PermCharInfo).
gap> infoperms:= PermCharInfo( tbl, perms );; gap> Display( tbl, infoperms.display ); M22 2 7 7 2 5 2 3 3 2 1 2 . 1 . 5 1 . . . . . 7 1 . . . . . 11 1 . . . . . 1a 2a 3a 4a 6a 8a 2P 1a 1a 3a 2a 3a 4a 3P 1a 2a 1a 4a 2a 8a 5P 1a 2a 3a 4a 6a 8a 7P 1a 2a 3a 4a 6a 8a 11P 1a 2a 3a 4a 6a 8a I.1 3080 56 2 12 2 2 I.2 1 21 8 54 24 36 I.3 1 3 4 9 12 18 I.4 3080 8 2 8 2 4 I.5 1 3 8 36 24 72 I.6 1 3 4 9 12 18 I.7 3080 24 11 4 3 2 I.8 1 9 44 18 36 36 I.9 1 3 4 9 12 18 I.10 3080 24 20 4 . 2 I.11 1 9 80 18 . 36 I.12 1 3 4 9 12 18
We have four candidates. For each the above list shows first the character values, then the cardinality of the intersection of \(U\) with the classes, and then lower bounds for the lengths of \(U\)-conjugacy classes of these elements. Only those classes of \(G\) are shown that contain elements of \(U\) for at least one of the characters.
If the first two candidates are permutation characters corresponding to \(U\) then \(U\) contains exactly \(8\) elements of order \(3\) and thus \(U\) has a normal Sylow \(3\) subgroup \(P\). But the order of \(N_G(P)\) is bounded by \(72\), which can be shown as follows. The only elements in \(G\) with centralizer order divisible by \(9\) are of order \(1\) or \(3\), so \(P\) is self-centralizing in \(G\). The factor \(N_G(P)/C_G(P)\) is isomorphic with a subgroup of Aut\((G) \cong GL(2,3)\) which has order divisible by \(16\), hence the order of \(N_G(P)\) divides \(144\). Now note that \([ G : N_G(P) ] \equiv 1 \pmod{3}\) by Sylow's Theorem, and \(|G|/144 = 3\,080 \equiv -1 \pmod{3}\). Thus the first two candidates are not permutation characters.
If the last two candidates are permutation characters corresponding to \(U\) then \(U\) has self-normalizing Sylow subgroups. This is because the index of a Sylow \(2\) normalizer in \(G\) is odd and divides \(9\), and if it is smaller than \(9\) then \(U\) contains at most \(3 \cdot 15 + 1\) elements of \(2\) power order; the index of a Sylow \(3\) normalizer in \(G\) is congruent to \(1\) modulo \(3\) and divides \(16\), and if it is smaller than \(16\) then \(U\) contains at most \(4 \cdot 8\) elements of order \(3\).
But since \(U\) is solvable and not a \(p\)-group, not all its Sylow subgroups can be self-normalizing; note that \(U\) has a proper normal subgroup \(N\) containing a Sylow \(p\) subgroup \(P\) of \(U\) for a prime divisor \(p\) of \(|U|\), and \(U = N \cdot N_U(P)\) holds by the Frattini argument (see [Hup67, Satz I.7.8]).
Let \(G\) be a maximal subgroup with structure \(3^{{2+4}}:2A_5.D_8\) in the sporadic simple Lyons group \(Ly\). We want to compute the permutation character \(1_G^{Ly}\). (This construction has been explained in [BP98, Section 4.2], without showing explicit GAP code.)
In the representation of \(Ly\) as automorphism group of the rank \(5\) graph B with \(9\,606\,125\) points (see [CCN+85, p. 174]), \(G\) is the stabilizer of an edge. A group \(S\) with structure \(3.McL.2\) is the point stabilizer. So the two point stabilizer \(U = S \cap G\) is a subgroup of index \(2\) in \(G\). The index of \(U\) in \(S\) is \(15\,400\), and according to the list of maximal subgroups of \(McL.2\) (see [CCN+85, p. 100]), the group \(U\) is isomorphic to the preimage in \(3.McL.2\) of a subgroup \(H\) of \(McL.2\) with structure \(3_+^{{1+4}}:4S_5\).
Using the improved combinatorial method described in [BP98, Section 3.2], all possible permutation characters of degree \(15\,400\) for the group \(McL\) are computed. (The method of [BP98, Section 3.3] is slower but also needs only a few seconds.)
gap> ly:= CharacterTable( "Ly" );; gap> mcl:= CharacterTable( "McL" );; gap> mcl2:= CharacterTable( "McL.2" );; gap> 3mcl2:= CharacterTable( "3.McL.2" );; gap> perms:= PermChars( mcl, rec( degree:= 15400 ) ); [ Character( CharacterTable( "McL" ), [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 0, 2, 1, 1, 1, 0, 0, 3, 0, 0, 1, 1, 1, 1 ] ), Character( CharacterTable( "McL" ), [ 15400, 280, 10, 37, 20, 0, 5, 10, 1, 0, 0, 2, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0 ] ) ]
We get two characters, corresponding to the two classes of maximal subgroups of index \(15\,400\) in \(McL\). The permutation character \(\pi = 1_{{H \cap McL}}^{McL}\) is the one with nonzero value on the class 10A, since the subgroup of structure \(2S_5\) in \(H \cap McL\) contains elements of order \(10\).
gap> ord10:= Filtered( [ 1 .. NrConjugacyClasses( mcl ) ], > i -> OrdersClassRepresentatives( mcl )[i] = 10 ); [ 15 ] gap> List( perms, pi -> pi[ ord10[1] ] ); [ 1, 0 ] gap> pi:= perms[1]; Character( CharacterTable( "McL" ), [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 0, 2, 1, 1, 1, 0, 0, 3, 0, 0, 1, 1, 1, 1 ] )
The character \(1_H^{McL.2}\) is an extension of \(\pi\), so we can use the method of [BP98, Section 3.3] to compute all possible permutation characters for the group \(McL.2\) that have the values of \(\pi\) on the classes of \(McL\). We find that the extension of \(\pi\) to a permutation character of \(McL.2\) is unique. Regarded as a character of \(3.McL.2\), this character is equal to \(1_U^S\).
gap> map:= InverseMap( GetFusionMap( mcl, mcl2 ) ); [ 1, 2, 3, 4, 5, 6, 7, 8, 9, [ 10, 11 ], 12, [ 13, 14 ], 15, 16, 17, 18, [ 19, 20 ], [ 21, 22 ], [ 23, 24 ] ] gap> torso:= CompositionMaps( pi, map ); [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 2, 1, 1, 0, 0, 3, 0, 1, 1 ] gap> perms:= PermChars( mcl2, rec( torso:= torso ) ); [ Character( CharacterTable( "McL.2" ), [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 2, 1, 1, 0, 0, 3, 0, 1, 1, 110, 26, 2, 4, 0, 0, 5, 2, 1, 1, 0, 0, 1, 1 ] ) ] gap> pi:= Inflated( perms[1], 3mcl2 ); Character( CharacterTable( "3.McL.2" ), [ 15400, 15400, 56, 56, 91, 91, 10, 12, 12, 25, 25, 0, 0, 11, 11, 2, 2, 0, 0, 0, 2, 2, 1, 1, 1, 0, 0, 0, 0, 3, 3, 0, 0, 0, 1, 1, 1, 1, 1, 1, 110, 26, 2, 4, 0, 0, 5, 2, 1, 1, 0, 0, 1, 1 ] )
The fusion of conjugacy classes of \(S\) in \(Ly\) can be computed from the character tables of \(S\) and \(Ly\) given in [CCN+85], it is unique up to Galois automorphisms of the table of \(Ly\).
gap> fus:= PossibleClassFusions( 3mcl2, ly );; Length( fus ); 4 gap> g:= AutomorphismsOfTable( ly );; gap> OrbitLengths( g, fus, OnTuples ); [ 4 ]
Now we can induce \(1_U^S\) to \(Ly\), which yields \((1_U^S)^{Ly} = 1_U^{Ly}\).
gap> pi:= Induced( 3mcl2, ly, [ pi ], fus[1] )[1]; Character( CharacterTable( "Ly" ), [ 147934325000, 286440, 1416800, 1082, 784, 12500, 0, 672, 42, 24, 0, 40, 0, 2, 20, 0, 0, 0, 64, 10, 0, 50, 2, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] )
All elements of odd order in \(G\) are contained in \(U\), for such an element \(g\) we have
\[ 1_G^{Ly}(g) = |C_{Ly}(g)| / |G| \cdot |G \cap Cl_{Ly}(g)| = |C_{Ly}(g)| / (2 \cdot |U|) \cdot |U \cap Cl_{Ly}(g)| = 1/2 \cdot 1_U^{Ly}(g) \ , \]
so we can prescribe the values of \(1_G^{Ly}\) on all classes of odd element order. For elements \(g\) of even order we have the weaker condition \(U\cap Cl_{Ly}(g) \subseteq G \cap Cl_{Ly}(g)\) and thus \(1_G^{Ly}(g) \geq 1/2 \cdot 1_U^{Ly}(g)\), which gives lower bounds for the value of \(1_G^{Ly}\) on the remaining classes.
gap> orders:= OrdersClassRepresentatives( ly ); [ 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, 14, 15, 15, 15, 18, 20, 21, 21, 22, 22, 24, 24, 24, 25, 28, 30, 30, 31, 31, 31, 31, 31, 33, 33, 37, 37, 40, 40, 42, 42, 67, 67, 67 ] gap> torso:= [];; gap> for i in [ 1 .. Length( orders ) ] do > if orders[i] mod 2 = 1 then > torso[i]:= pi[i]/2; > fi; > od; gap> torso; [ 73967162500,, 708400, 541,, 6250, 0,,,, 0,,, 1,,, 0, 0,,,, 25, 1, 0, ,, 0, 0,,,,,, 0,,,, 0, 0, 0, 0, 0, 0, 0, 0, 0,,,,, 0, 0, 0 ]
Exactly one possible permutation character of \(Ly\) satisfies these conditions.
gap> perms:= PermChars( ly, rec( torso:= torso ) );; gap> Length( perms ); 43 gap> perms:= Filtered( perms, cand -> ForAll( [ 1 .. Length( orders ) ], > i -> cand[i] >= pi[i] / 2 ) ); [ Character( CharacterTable( "Ly" ), [ 73967162500, 204820, 708400, 541, 392, 6250, 0, 1456, 61, 25, 0, 22, 10, 1, 10, 0, 0, 0, 32, 5, 0, 25, 1, 0, 1, 2, 0, 0, 0, 0, 4, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0 ] ) ]
(The permutation character \(1_G^{Ly}\) was used in the proof that the character \(\chi_{37}\) of \(Ly\) (see [CCN+85, p. 175]) occurs with multiplicity at least 2 in each character of \(Ly\) that is induced from a proper subgroup of \(Ly\).)
According to the Atlas of Finite Groups [CCN+85, p. 34], the group Aut\((U_3(5))\) has two classes of maximal subgroups of order \(2^4 \cdot 3^3\), which have the structures \(3^2 \colon 2S_4\) and \(6^2 \colon D_{12}\), respectively.
gap> tbl:= CharacterTable( "U3(5).3.2" ); CharacterTable( "U3(5).3.2" ) gap> deg:= Size( tbl ) / ( 2^4*3^3 ); 1750 gap> pi:= PermChars( tbl, rec( torso:= [ deg ] ) ); [ Character( CharacterTable( "U3(5).3.2" ), [ 1750, 70, 13, 2, 0, 0, 1, 0, 0, 0, 10, 7, 10, 4, 2, 0, 0, 0, 0, 0, 0, 30, 10, 3, 0, 0, 1, 0, 0 ] ), Character( CharacterTable( "U3(5).3.2" ), [ 1750, 30, 4, 6, 0, 0, 0, 0, 0, 0, 40, 7, 0, 6, 0, 0, 0, 0, 0, 0, 0, 20, 0, 2, 2, 0, 0, 0, 0 ] ) ]
Now the question is which character belongs to which subgroup. We see that the first character vanishes on the classes of element order \(8\) and the second does not, so only the first one can be the permutation character induced from \(6^2 \colon D_{12}\).
gap> ord8:= Filtered( [ 1 .. NrConjugacyClasses( tbl ) ], > i -> OrdersClassRepresentatives( tbl )[i] = 8 ); [ 9, 25 ] gap> List( pi, x -> x{ ord8 } ); [ [ 0, 0 ], [ 0, 2 ] ]
Thus the question is whether the second candidate is really a permutation character. Since none of the two candidates vanishes on any outer coset of \(U_3(5)\) in Aut\((U_3(5))\), the point stabilizers are extensions of groups of order \(2^3 \cdot 3^2\) in \(U_3(5)\). The restrictions of the candidates to \(U_3(5)\) are different, so we can try to answer the question using information about this group.
gap> subtbl:= CharacterTable( "U3(5)" ); CharacterTable( "U3(5)" ) gap> rest:= RestrictedClassFunctions( pi, subtbl ); [ Character( CharacterTable( "U3(5)" ), [ 1750, 70, 13, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U3(5)" ), [ 1750, 30, 4, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
The intersection of the \(3^2 \colon 2S_4\) subgroup with \(U_3(5)\) lies inside the maximal subgroup of type \(M_{10}\), which does not contain elements of order\(6\). Only the second character has this property.
gap> ord6:= Filtered( [ 1 .. NrConjugacyClasses( subtbl ) ], > i -> OrdersClassRepresentatives( subtbl )[i] = 6 ); [ 9 ] gap> List( rest, x -> x{ ord6 } ); [ [ 1 ], [ 0 ] ]
In order to establish the two characters as permutation characters, we could also compute the permutation characters of the degree in question directly from the table of marks of \(U_3(5)\), which is contained in the GAP library of tables of marks.
gap> tom:= TableOfMarks( "U3(5)" ); TableOfMarks( "U3(5)" ) gap> perms:= PermCharsTom( subtbl, tom );; gap> Set( Filtered( perms, x -> x[1] = deg ) ) = Set( rest ); true
We were mainly interested in the multiplicities of irreducible characters in these characters. The action of Aut\((U_3(5)\) on the cosets of \(3^2 \colon 2S_4\) turns out to be multiplicity-free whereas that on the cosets of \(6^2 \colon D_{12}\) is not.
gap> PermCharInfo( tbl, pi ).ATLAS; [ "1a+21a+42a+84aac+105a+125a+126a+250a+252a+288bc", "1a+42a+84ac+105ab+125a+126a+250a+252b+288bc" ]
It should be noted that the restrictions of the multiplicity-free character to the subgroups \(U_3(5).2\) and \(U_3(5).3\) of Aut\((U_3(5)\) are not multiplicity-free.
gap> subtbl2:= CharacterTable( "U3(5).2" );; gap> rest2:= RestrictedClassFunctions( pi, subtbl2 );; gap> PermCharInfo( subtbl2, rest2 ).ATLAS; [ "1a+21aab+28aa+56aa+84a+105a+125aab+126aab+288aa", "1a+21ab+28a+56a+84a+105ab+125aab+126a+252a+288aa" ] gap> subtbl3:= CharacterTable( "U3(5).3" );; gap> rest3:= RestrictedClassFunctions( pi, subtbl3 );; gap> PermCharInfo( subtbl3, rest3 ).ATLAS; [ "1a+21abc+84aab+105a+125abc+126abc+144bcef", "1a+21bc+84ab+105aa+125abc+126adg+144bcef" ]
According to the Atlas of Finite Groups [CCN+85, p. 85], the group \(G =\) Aut\((O_8^+(2))\) has a class of maximal subgroups of order \(2^{13} \cdot 3^2\), thus the index of these subgroups in \(G\) is \(3^4 \cdot 5^2 \cdot 7\). The intersection of these subgroups with \(H = O_8^+(2)\) lie inside maximal subgroups of type \(2^6 \colon A_8\). We want to show that the permutation character of the action of \(G\) on the cosets of these subgroups is not multiplicity-free.
Since the table of marks for \(H\) is available in GAP, but not that for \(G\), we first compute the \(H\)-permutation characters of the intersections with \(H\) of index \(3^4 \cdot 5^2 \cdot 7 = 14\,175\) subgroups in \(G\).
(Note that these intersections have order \(2^{12} \cdot 3\) because subgroups of order \(2^{12} \cdot 3^2\) are contained in \(O_8^+(2).2\) and hence are not maximal in \(G\).)
gap> t:= CharacterTable( "O8+(2).3.2" );; gap> s:= CharacterTable( "O8+(2)" );; gap> tom:= TableOfMarks( s );; gap> perms:= PermCharsTom( s, tom );; gap> deg:= 3^4*5^2*7; 14175 gap> perms:= Filtered( perms, x -> x[1] = deg );; gap> Length( perms ); 4 gap> Length( Set( perms ) ); 1
We see that there are four classes of subgroups \(S\) in \(H\) that may belong to maximal subgroups of the desired index in \(G\), and that the permutation characters are equal. They lead to such groups if they extend to \(G\), so we compute the possible permutation characters of \(G\) that extend these characters.
gap> fus:= PossibleClassFusions( s, t ); [ [ 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 8, 9, 10, 10, 10, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 19, 20, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 26, 26, 26, 27, 27, 27 ] ] gap> fus:= fus[1];; gap> inv:= InverseMap( fus );; gap> comp:= CompositionMaps( perms[1], inv ); [ 14175, 1215, 375, 79, 0, 0, 27, 27, 99, 15, 7, 0, 0, 0, 0, 9, 3, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0 ] gap> ext:= PermChars( t, rec( torso:= comp ) ); [ Character( CharacterTable( "O8+(2).3.2" ), [ 14175, 1215, 375, 79, 0, 0, 27, 27, 99, 15, 7, 0, 0, 0, 0, 9, 3, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 63, 9, 15, 7, 1, 0, 3, 3, 3, 1, 0, 0, 1, 1, 945, 129, 45, 69, 21, 25, 13, 0, 0, 0, 9, 0, 3, 3, 7, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0 ] ) ] gap> PermCharInfo( t, ext[1] ).ATLAS; [ "1a+50b+100a+252bb+300b+700b+972bb+1400a+1944a+3200b+4032b" ]
Thus we get one permutation character of \(G\) which is not multiplicity-free.
In this section, we compute four primitive permutation characters \(1_H^M\) of the sporadic simple Monster group \(M\), using the following strategy.
Let \(E\) be an elementary abelian \(2\)-subgroup of \(M\), and \(H = N_M(E)\). For an involution \(z \in E\), let \(G = C_M(z)\) and \(U = G \cap H = C_H(z)\) and \(V = C_H(E)\), a normal subgroup of \(H\). According to the Atlas of Finite Groups [CCN+85, p. 234], \(G\) has the structure \(2.B\) if \(z\) is in the class 2A of \(M\), and \(G\) has the structure \(2^{{1+24}}_+.Co_1\) if \(z\) is in the class 2B of \(M\). In the latter case, let \(N\) denote the extraspecial normal subgroup of order \(2^{25}\) in \(G\). It will turn out that in our situation, \(U\) contains \(N\).
We want to compute many values of \(1_H^M\) from the knowledge of permutation characters \(1_X^M\), for suitable subgroups \(X\) with the property \(V \leq X \leq U\), and then use the GAP function PermChars (Reference: PermChars) for computing all those possible permutation characters of \(M\) that take the known values; if there is a unique solution then this is the desired character \(1_H^M\).
Why does this approach have a chance to be successful? Currently we do not have representations for the subgroups \(H\) in question, but the character tables of the involution centralizers \(G\) in \(M\) are available, and also either the character tables of \(X/V\) for the interesting subgroups \(X\) are known or we have enough information to compute the characters \(1_X^G\).
And how do we compute certain values of \(1_H^M\)? Suppose that \(C\) is a union of classes of \(M\) and \(I\) is an index set such that \((1_H)_{{C \cap H}} = (\sum_{{i \in I}} c_i 1_{{X_i}}^H)_{{C \cap H}}\) holds for suitable rational numbers \(c_i\).
The right hand side of this equality lives in \(H/V\), provided that \(C\) "behaves well" w.r.t. factoring out the normal subgroup \(V\) of \(H\), i. e., if there is a set of classes in \(H/V\) whose preimages in \(H\) form the set \(H \cap C\). For example, \(C\) may be the set of all those elements in \(M\) whose order is not divisible by a particular prime \(p\) that divides \(|H|\) but not \(|U|\).
Under these conditions, we have \((1_H^M)_{C} = ((\sum_{{i \in I}} c_i 1_{{X_i}}^G)^M)_{C}\), and we interpret the right hand side as follows: If \(X_i\) contains \(N\) then \(1_{{X_i}}^G\) can be identified with \(1_{{X_i/N}}^{{G/N}}\). If \(X_i\) contains at least \(Z\) then \(1_{{X_i}}^G\) can be identified with \(1_{{X_i/Z}}^{{G/Z}}\). As mentioned above, we have good chances to compute these characters. So the main task in each of the following sections is to find, for a suitable set \(C\) of classes, a linear combination of permutation characters of \(H/V\) whose restriction to \((C \cap H) / V\) is constant and nonzero.
According to the Atlas of Finite Groups [CCN+85, p. 234], the Monster group \(M\) has a class of maximal subgroups \(H\) of the type \(2^2.2^{11}.2^{22}.(S_3 \times M_{24})\). Currently the character table of \(H\) and the class fusion into \(M\) are not available in GAP. We are interested in the permutation character \(1_H^G\), and we will compute it without this information.
The subgroup \(H\) normalizes a Klein four group \(E\) whose involutions lie in the class 2B. We fix an involution \(z\) in \(E\), and set \(G = C_M(z)\), \(U = C_H(z)\), and \(V = C_H(E)\). Further, let \(N\) be the extraspecial normal subgroup of order \(2^{25}\) in \(G\).
So \(G\) has the structure \(2^{{1+24}}_+.Co_1\), and \(U\) has index three in \(H\). The order of \(N U / N\) is a multiple of \(2^{{2+11+22-25}} \cdot 2 \cdot |M_{24}|\), and \(N U / N\) occurs as a subgroup of \(G / N \cong Co_1\).
gap> co1:= CharacterTable( "Co1" );; gap> order:= 2^(2+11+22-25) * 2 * Size( CharacterTable( "M24" ) ); 501397585920 gap> maxes:= List( Maxes( co1 ), CharacterTable );; gap> filt:= Filtered( maxes, t -> Size( t ) mod order = 0 ); [ CharacterTable( "2^11:M24" ) ] gap> List( filt, t -> Size( t ) / order ); [ 1 ] gap> k:= filt[1];;
The list of maximal subgroups of \(Co_1\) (see [CCN+85, p. 183]) tells us that \(NU / N\) is a maximal subgroup \(K\) of \(Co_1\) and has the structure \(2^{11}:M_{24}\). In particular, \(U\) contains \(N\) and thus \(U/N \cong K\).
Let \(C = \{ g \in M; 3 \nmid |g|\) or \(1_V^M(g^3) = 0 \}\).
Then \((1_H)_{{C \cap H}} = (1_U^H - 1/3 1_V^H)_{{C \cap H}}\) holds, as we can see from computations with \(H/V \cong S_3\), as follows.
gap> f:= CharacterTable( "Symmetric", 3 ); CharacterTable( "Sym(3)" ) gap> OrdersClassRepresentatives( f ); [ 1, 2, 3 ] gap> deg3:= PermChars( f, 3 ); [ Character( CharacterTable( "Sym(3)" ), [ 3, 1, 0 ] ) ] gap> deg6:= PermChars( f, 6 ); [ Character( CharacterTable( "Sym(3)" ), [ 6, 0, 0 ] ) ] gap> deg3[1] - 1/3 * deg6[1]; ClassFunction( CharacterTable( "Sym(3)" ), [ 1, 1, 0 ] )
The character table of \(G\) is available in GAP, so we can compute the permutation character \(\pi = 1_U^G\) by computing the primitive permutation character \(1_K^{{Co_1}}\), identifying it with \(1_{{U/N}}^{{G/N}}\), and then inflating this character to \(G\).
gap> m:= CharacterTable( "M" ); CharacterTable( "M" ) gap> g:= CharacterTable( "MC2B" ); CharacterTable( "2^1+24.Co1" ) gap> pi:= RestrictedClassFunction( TrivialCharacter( k )^co1, g );;
Next we consider the permutation character \(\phi = 1_V^G\). The group \(V\) does not contain \(N\) because \(K\) is perfect. But \(V\) contains \(Z\) because otherwise \(U\) would be a direct product of \(V\) and \(Z\), which would imply that \(N\) would be a direct product of \(V \cap N\) and \(Z\). So we can regard \(\phi\) as the inflation of \(1_{{V/Z}}^{{G/Z}}\) from \(G/Z\) to \(G\), i. e., we can perform the computations with the character table of the factor group \(G/Z\).
gap> zclasses:= ClassPositionsOfCentre( g );; gap> gmodz:= g / zclasses; CharacterTable( "2^1+24.Co1/[ 1, 2 ]" ) gap> invmap:= InverseMap( GetFusionMap( g, gmodz ) );; gap> pibar:= CompositionMaps( pi, invmap );;
Since \(\phi(g) = [G:V] \cdot |g^G \cap V| / |g^G|\) holds for \(g \in G\), and since \(g^G \cap V \subseteq g^G \cap VN\), with equality if \(g\) has odd order, we get \(\phi(g) = 2 \cdot \pi(g)\) if \(g\) has odd order, and \(\phi(g) = 0\) if \(\pi(g) = 0\).
We want to compute the possible permutation characters with these values.
gap> factorders:= OrdersClassRepresentatives( gmodz );; gap> phibar:= [];; gap> for i in [ 1 .. NrConjugacyClasses( gmodz ) ] do > if factorders[i] mod 2 = 1 then > phibar[i]:= 2 * pibar[i]; > elif pibar[i] = 0 then > phibar[i]:= 0; > fi; > od; gap> cand:= PermChars( gmodz, rec( torso:= phibar ) );; gap> Length( cand ); 1
Now we know \(\pi^M = 1_U^M\) and \(\phi^M = 1_V^M\), so we can write down \((1_H^M)_{C}\).
gap> phi:= RestrictedClassFunction( cand[1], g )^m;; gap> pi:= pi^m;; gap> cand:= ShallowCopy( pi - 1/3 * phi );; gap> morders:= OrdersClassRepresentatives( m );; gap> for i in [ 1 .. Length( morders ) ] do > if morders[i] mod 3 = 0 and phi[ PowerMap( m, 3 )[i] ] <> 0 then > Unbind( cand[i] ); > fi; > od;
We claim that \(1_H^M(g) \geq \pi^M(g) - 1/3 \psi^M(g)\) for all \(g \in M\). In order to see this, let \(H'\) denote the index two subgroup of \(H\), and let \(g \in M\). Since \(H\) is the disjoint union of \(V\), \(H' \setminus V\), and three \(H\)-conjugates of \(U \setminus V\), we get
| 1_H^M(g) | = | [M:H] ⋅ |g^M ∩ H| / |g^M| |
| = | [M:H] ⋅ ( |g^M ∩ V| + 3 |g^M ∩ U \ V| + |g^M ∩ H' \ V| ) / |g^M| | |
| = | [M:H] ⋅ ( 3 |g^M ∩ U| - 2 |g^M ∩ V| + |g^M ∩ H' \ V| ) / |g^M| | |
| = | 1_U^M(g) - 1/3 ⋅ 1_V^G(g) + [M:H] ⋅ |g^M ∩ H' \ V| / |g^M| . |
Possible constituents of \(1_H^M\) are those rational irreducible characters of \(M\) that are constituents of \(\pi^M\).
gap> constit:= Filtered( RationalizedMat( Irr( m ) ), > chi -> ScalarProduct( m, chi, pi ) <> 0 );;
Now we compute the possible permutation characters that have the prescribed values, are compatible with the given lower bounds for values, and have only constituents in the given list.
gap> cand:= PermChars( m, > rec( torso:= cand, chars:= constit, > lower:= ShallowCopy( pi - 1/3 * phi ), > normalsubgroup:= [ 1 .. NrConjugacyClasses( m ) ], > nonfaithful:= TrivialCharacter( m ) ) ); [ Character( CharacterTable( "M" ), [ 16009115629875684006343550944921875, 7774182899642733721875, 120168544413337875, 4436049512692980, 215448838605, 131873639625, 760550656275, 110042727795, 943894035, 568854195, 1851609375, 0, 4680311220, 405405, 78624756, 14467005, 178605, 248265, 874650, 0, 76995, 591163, 224055, 34955, 29539, 20727, 0, 0, 375375, 15775, 0, 0, 0, 495, 116532, 3645, 62316, 1017, 11268, 357, 1701, 45, 117, 705, 0, 0, 4410, 1498, 0, 3780, 810, 0, 0, 83, 135, 31, 0, 0, 0, 0, 0, 0, 0, 255, 195, 0, 215, 0, 0, 210, 0, 42, 0, 35, 15, 1, 1, 160, 48, 9, 92, 25, 9, 9, 5, 1, 21, 0, 0, 0, 0, 0, 98, 74, 42, 0, 0, 0, 120, 76, 10, 0, 0, 0, 0, 0, 1, 1, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 3, 0, 0, 0, 18, 0, 10, 0, 3, 3, 0, 1, 1, 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 12, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
There is only one candidate, so we have found the permutation character.
According to the Atlas of Finite Groups [CCN+85, p. 234], the Monster group \(M\) has a class of maximal subgroups \(H\) of the type \(2^3.2^6.2^{12}.2^{18}.(L_3(2) \times 3.S_6)\). Currently the character table of \(H\) and the class fusion into \(M\) are not available in GAP. We are interested in the permutation character \(1_H^G\), and we will compute it without this information.
The subgroup \(H\) normalizes an elementary abelian group \(E\) of order eight whose involutions lie in the class 2B. We fix an involution \(z\) in \(E\), and set \(G = C_M(z)\), \(U = C_H(z)\), and \(V = C_H(E)\). Further, let \(N\) be the extraspecial normal subgroup of order \(2^{25}\) in \(G\).
So \(G\) has the structure \(2^{{1+24}}_+.Co_1\), and \(U\) has index seven in \(H\). The order of \(N U / N\) is a multiple of \(2^{{3+6+12+18-25}} \cdot |L_3(2)| \cdot |3.S_6| / 7\), and \(N U / N\) occurs as a subgroup of \(G / N \cong Co_1\).
gap> co1:= CharacterTable( "Co1" );; gap> order:= 2^(3+6+12+18-25) * 168 * 3 * Factorial( 6 ) / 7; 849346560 gap> maxes:= List( Maxes( co1 ), CharacterTable );; gap> filt:= Filtered( maxes, t -> Size( t ) mod order = 0 ); [ CharacterTable( "2^(1+8)+.O8+(2)" ), CharacterTable( "2^(4+12).(S3x3S6)" ) ] gap> List( filt, t -> Size( t ) / order ); [ 105, 1 ] gap> o8p2:= CharacterTable( "O8+(2)" );; gap> PermChars( o8p2, rec( torso:= [ 105 ] ) ); [ ] gap> k:= filt[2];;
The list of maximal subgroups of \(Co_1\) (see [CCN+85, p. 183]) tells us that \(NU / N\) is a maximal subgroup \(K\) of \(Co_1\) and has the structure \(2^{{4+12}}.(S_3 \times 3.S_6)\). (Note that the group \(O_8^+(2)\) has no proper subgroup of index \(105\).) In particular, \(U\) contains \(N\) and thus \(U/N \cong K\).
Let \(C\) be the set of elements in \(M\) whose order is not divisible by \(7\). Then \((1_H)_{{C \cap H}} = (1_U^H - 1/3 1_{VN}^H + 1/21 1_V^H)_{{C \cap H}}\) holds, as we can see from computations with \(H/V \cong L_3(2)\), as follows.
So S4, V4, 1 suffice! -->
gap> f:= CharacterTable( "L3(2)" ); CharacterTable( "L3(2)" ) gap> OrdersClassRepresentatives( f ); [ 1, 2, 3, 4, 7, 7 ] gap> deg7:= PermChars( f, 7 ); [ Character( CharacterTable( "L3(2)" ), [ 7, 3, 1, 1, 0, 0 ] ) ] gap> deg42:= PermChars( f, 42 ); [ Character( CharacterTable( "L3(2)" ), [ 42, 2, 0, 2, 0, 0 ] ), Character( CharacterTable( "L3(2)" ), [ 42, 6, 0, 0, 0, 0 ] ) ] gap> deg168:= PermChars( f, 168 ); [ Character( CharacterTable( "L3(2)" ), [ 168, 0, 0, 0, 0, 0 ] ) ] gap> deg7[1] - 1/3 * deg42[2] + 1/21 * deg168[1]; ClassFunction( CharacterTable( "L3(2)" ), [ 1, 1, 1, 1, 0, 0 ] )
(Note that \(VN/V\) is a Klein four group, and there is only one transitive permutation character of \(L_3(2)\) that is induced from such subgroups.)
The character table of \(G\) is available in GAP, so we can compute the permutation character \(\pi = 1_U^G\) by computing the primitive permutation character \(1_K^{{Co_1}}\), identifying it with \(1_{{U/N}}^{{G/N}}\), and then inflating this character to \(G\).
gap> m:= CharacterTable( "M" ); CharacterTable( "M" ) gap> g:= CharacterTable( "MC2B" ); CharacterTable( "2^1+24.Co1" ) gap> pi:= RestrictedClassFunction( TrivialCharacter( k )^co1, g );;
The permutation character \(\psi = 1_{VN}^G\) can be computed as the inflation of \(1_{{VN/N}}^{{G/N}} = (1_{{VN/N}}^{{U/N}})^{{G/N}}\), where \(1_{{VN/N}}^{{U/N}}\) is a character of \(K\) that can be identified with the regular permutation character of \(U/VN \cong S_3\).
gap> nsg:= ClassPositionsOfNormalSubgroups( k );; gap> nsgsizes:= List( nsg, x -> Sum( SizesConjugacyClasses( k ){ x } ) );; gap> nn:= nsg[ Position( nsgsizes, Size( k ) / 6 ) ];; gap> psi:= 0 * [ 1 .. NrConjugacyClasses( k ) ];; gap> for i in nn do > psi[i]:= 6; > od; gap> psi:= InducedClassFunction( k, psi, co1 );; gap> psi:= RestrictedClassFunction( psi, g );;
Next we consider the permutation character \(\phi = 1_V^G\). The group \(V\) does not contain \(N\) because \(K\) does not have a factor group of the type \(S_4\). But \(V\) contains \(Z\) because \(U/V\) is centerless. So we can regard \(\phi\) as the inflation of \(1_{{V/Z}}^{{G/Z}}\) from \(G/Z\) to \(G\), i. e., we can perform the computations with the character table of the factor group \(G/Z\).
gap> zclasses:= ClassPositionsOfCentre( g );; gap> gmodz:= g / zclasses; CharacterTable( "2^1+24.Co1/[ 1, 2 ]" ) gap> invmap:= InverseMap( GetFusionMap( g, gmodz ) );; gap> psibar:= CompositionMaps( psi, invmap );;
Since \(\phi(g) = [G:V] \cdot |g^G \cap V| / |g^G|\) holds for \(g \in G\), and since \(g^G \cap V \subseteq g^G \cap VN\), with equality if \(g\) has odd order, we get \(\phi(g) = 4 \cdot \psi(g)\) if \(g\) has odd order, and \(\phi(g) = 0\) if \(\psi(g) = 0\).
We want to compute the possible permutation characters with these values. This is easier if we "go down" from \(VN\) to \(V\) in two steps.
gap> factorders:= OrdersClassRepresentatives( gmodz );; gap> phibar:= [];; gap> upperphibar:= [];; gap> for i in [ 1 .. NrConjugacyClasses( gmodz ) ] do > if factorders[i] mod 2 = 1 then > phibar[i]:= 2 * psibar[i]; > elif psibar[i] = 0 then > phibar[i]:= 0; > fi; > upperphibar[i]:= 2 * psibar[i]; > od; gap> cand:= PermChars( gmodz, rec( torso:= phibar, > upper:= upperphibar, > normalsubgroup:= [ 1 .. NrConjugacyClasses( gmodz ) ], > nonfaithful:= TrivialCharacter( gmodz ) ) );; gap> Length( cand ); 3
One of the candidates computed in this first step is excluded by the fact that it is induced from a subgroup that contains \(N/Z\).
gap> nn:= First( ClassPositionsOfNormalSubgroups( gmodz ), > x -> Sum( SizesConjugacyClasses( gmodz ){x} ) = 2^24 ); [ 1 .. 4 ] gap> cont:= PermCharInfo( gmodz, cand ).contained;; gap> cand:= cand{ Filtered( [ 1 .. Length( cand ) ], > i -> Sum( cont[i]{ nn } ) < 2^24 ) };; gap> Length( cand ); 2
Now we run the second step. After excluding the candidates that cannot be induced from subgroups whose intersection with \(N/Z\) has index four in \(N/Z\), we get four solutions.
gap> poss:= [];; gap> for v in cand do > phibar:= []; > upperphibar:= []; > for i in [ 1 .. NrConjugacyClasses( gmodz ) ] do > if factorders[i] mod 2 = 1 then > phibar[i]:= 2 * v[i]; > elif v[i] = 0 then > phibar[i]:= 0; > fi; > upperphibar[i]:= 2 * v[i]; > od; > Append( poss, PermChars( gmodz, rec( torso:= phibar, > upper:= upperphibar, > normalsubgroup:= [ 1 .. NrConjugacyClasses( gmodz ) ], > nonfaithful:= TrivialCharacter( gmodz ) ) ) ); > od; gap> Length( poss ); 6 gap> cont:= PermCharInfo( gmodz, poss ).contained;; gap> poss:= poss{ Filtered( [ 1 .. Length( poss ) ], > i -> Sum( cont[i]{ nn } ) < 2^23 ) };; gap> Length( poss ); 4 gap> phicand:= RestrictedClassFunctions( poss, g );;
Since we have several candidates for \(1_V^G\), we form the linear combinations for all these candidates.
gap> phicand:= RestrictedClassFunctions( poss, g );; gap> phicand:= InducedClassFunctions( phicand, m );; gap> psi:= psi^m;; gap> pi:= pi^m;; gap> cand:= List( phicand, > phi -> ShallowCopy( pi - 1/3 * psi + 1/21 * phi ) );; gap> morders:= OrdersClassRepresentatives( m );; gap> for x in cand do > for i in [ 1 .. Length( morders ) ] do > if morders[i] mod 7 = 0 then > Unbind( x[i] ); > fi; > od; > od;
Exactly one of the candidates has only integral values.
gap> cand:= Filtered( cand, x -> ForAll( x, IsInt ) ); [ [ 4050306254358548053604918389065234375, 148844831270071996434375, 2815847622206994375, 14567365753025085, 3447181417680, 659368198125, 3520153823175, 548464353255, 5706077895, 3056566695, 264515625, 0, 19572895485, 6486480, 186109245, 61410960, 758160, 688365,,, 172503, 1264351, 376155, 137935, 99127, 52731, 0, 0, 119625, 3625, 0, 0, 0, 0, 402813, 29160, 185301, 2781, 21069, 1932, 4212, 360, 576, 1125, 0, 0,,,, 2160, 810, 0, 0, 111, 179, 43, 0, 0, 0, 0, 0, 0, 0, 185, 105, 0, 65, 0, 0,,,,, 0, 0, 0, 0, 337, 105, 36, 157, 37, 18, 18, 16, 4, 21, 0, 0, 0, 0, 0,,,,, 0, 0, 60, 40, 10, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0,,,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0,,,, 0, 0, 0, 6, 8, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0,,, 0, 0, 0, 0, 0,,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0 ] ]
Possible constituents of \(1_H^M\) are those rational irreducible characters of \(M\) that are constituents of \(\pi^M\).
gap> constit:= Filtered( RationalizedMat( Irr( m ) ), > chi -> ScalarProduct( m, chi, pi ) <> 0 );;
Now we compute the possible permutation characters that have the prescribed values and have only constituents in the given list.
gap> cand:= PermChars( m, rec( torso:= cand[1], chars:= constit ) ); [ Character( CharacterTable( "M" ), [ 4050306254358548053604918389065234375, 148844831270071996434375, 2815847622206994375, 14567365753025085, 3447181417680, 659368198125, 3520153823175, 548464353255, 5706077895, 3056566695, 264515625, 0, 19572895485, 6486480, 186109245, 61410960, 758160, 688365, 58310, 0, 172503, 1264351, 376155, 137935, 99127, 52731, 0, 0, 119625, 3625, 0, 0, 0, 0, 402813, 29160, 185301, 2781, 21069, 1932, 4212, 360, 576, 1125, 0, 0, 1302, 294, 0, 2160, 810, 0, 0, 111, 179, 43, 0, 0, 0, 0, 0, 0, 0, 185, 105, 0, 65, 0, 0, 224, 0, 14, 0, 0, 0, 0, 0, 337, 105, 36, 157, 37, 18, 18, 16, 4, 21, 0, 0, 0, 0, 0, 70, 38, 14, 0, 0, 0, 60, 40, 10, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0, 24, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 8, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0 ] ) ]
There is only one candidate, so we have found the permutation character.
According to the Atlas of Finite Groups [CCN+85, p. 234], the Monster group \(M\) has a class of maximal subgroups \(H\) of the type \(2^5.2^{10}.2^{20}.(S_3 \times L_5(2))\). Currently the character table of \(H\) and the class fusion into \(M\) are not available in GAP. We are interested in the permutation character \(1_H^G\), and we will compute it without this information.
The subgroup \(H\) normalizes an elementary abelian group \(E\) of order \(32\) whose involutions lie in the class 2B. We fix an involution \(z\) in \(E\), and set \(G = C_M(z)\), \(U = C_H(z)\), and \(V = C_H(E)\). Further, let \(N\) be the extraspecial normal subgroup of order \(2^{25}\) in \(G\).
So \(G\) has the structure \(2^{{1+24}}_+.Co_1\), and \(U\) has index \(31\) in \(H\). The order of \(N U / N\) is a multiple of \(2^{{5+10+20-25}} \cdot |L_5(2)| \cdot |S_3| / 31\), and \(N U / N\) occurs as a subgroup of \(G / N \cong Co_1\).
gap> co1:= CharacterTable( "Co1" );; gap> order:= 2^35*Size( CharacterTable( "L5(2)" ) )*6 / 2^25 / 31; 1981808640 gap> maxes:= List( Maxes( co1 ), CharacterTable );; gap> filt:= Filtered( maxes, t -> Size( t ) mod order = 0 ); [ CharacterTable( "2^11:M24" ), CharacterTable( "2^(1+8)+.O8+(2)" ), CharacterTable( "2^(2+12):(A8xS3)" ) ] gap> List( filt, t -> Size( t ) / order ); [ 253, 45, 1 ] gap> m24:= CharacterTable( "M24" );; gap> cand:= PermChars( m24, rec( torso:=[ 253 ] ) ); [ Character( CharacterTable( "M24" ), [ 253, 29, 13, 10, 1, 5, 5, 1, 3, 2, 1, 1, 1, 1, 3, 0, 2, 1, 1, 1, 0, 0, 1, 1, 0, 0 ] ) ] gap> TestPerm5( m24, cand, m24 mod 11 ); [ ] gap> PermChars( CharacterTable( "O8+(2)" ), rec( torso:=[ 45 ] ) ); [ ] gap> k:= filt[3];;
The list of maximal subgroups of \(Co_1\) (see [CCN+85, p. 183]) tells us that \(NU / N\) is a maximal subgroup \(K\) of \(Co_1\) and has the structure \(2^{{2+12}}.(A_8 \times S_3)\). (Note that the group \(M_{24}\) has no proper subgroup of index \(253\), which is shown above using the \(11\)-modular Brauer table of \(M_{24}\). Furthermore, the group \(O_8^+(2)\) has no subgroup of index \(45\).) In particular, \(U\) contains \(N\) and thus \(U/N \cong K\).
Let \(C\) be the set of elements in \(M\) whose order is not divisible by \(31\) or \(21\). We want to find an index set \(I\) and subgroups \(X_i\), for \(i \in I\), with the property that \(V \leq X_i \leq U\) and
\[ (1_H)_{{C \cap H}} = \left( \sum_{{i \in I}} c_i 1_{{X_i}}^H \right)_{{C \cap H}} \]
holds for suitable rational integers \(c_i\). Let \(W\) be the full preimage of the elementary normal subgroup of order \(16\) in \(U/V \cong 2^4.A_8\) under the natural epimorphism from \(U\) to \(U/V\), and set \(I_1 = \{ i \in I; W \leq X_i \}\) and \(I_2 = I \setminus I_1\).
Using the known table of marks of \(U/V\), we will find a solution such that \([W:(W \cap X_i)] = 2\) for all \(i \in I_2\). First we compute the permutation characters \(1_S^{{U/V}}\) for all subgroups \(S\) of \(U/V\) that contain \(W/V\), and induce them to \(H/V\).
gap> subtbl:= CharacterTable( "2^4:A8" );; gap> subtom:= TableOfMarks( subtbl );; gap> perms:= PermCharsTom( subtbl, subtom );; gap> nsg:= ClassPositionsOfNormalSubgroups( subtbl ); [ [ 1 ], [ 1, 2 ], [ 1 .. 25 ] ] gap> above:= Filtered( perms, x -> x[1] = x[2] );; gap> tbl:= CharacterTable( "L5(2)" );; gap> above:= Set( Induced( subtbl, tbl, above ) );;
Next we compute the permutation characters \(1_S^{{U/V}}\) for all subgroups \(S\) of \(U/V\) whose intersection with \(W/V\) has index two in \(W/V\). Afterwards we exclude certain subgroups that would slow down later computations, and induce also these characters to \(H/V\).
gap> index2:= Filtered( perms, > x -> Sum( PermCharInfo( subtbl, [x] ).contained[1]{ [1,2] } ) = 8 );; gap> index2:= Filtered( index2, x -> not x[1] in [ 630, 840, 1260, 1680 ] );; gap> index2:= Set( Induced( subtbl, tbl, index2 ) );;
Now we induce the permutation characters to \(H/V\), and compute the coefficients of a linear combination as desired.
gap> orders:= OrdersClassRepresentatives( tbl );; gap> goodclasses:= Filtered( [ 1 .. NrConjugacyClasses( tbl ) ], > i -> not orders[i] in [ 21, 31 ] ); [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ] gap> matrix:= List( Concatenation( above, index2 ), x -> x{ goodclasses } );; gap> sol:= SolutionMat( matrix, > ListWithIdenticalEntries( Length( goodclasses ), 1 ) ); [ 692/651, 57/217, -78/217, -26/217, 0, 74/651, 11/217, 0, 3/217, 151/651, 0, 22/651, 0, 0, 0, -11/217, 0, 0, 0, 0, 0, 0, 0, 0, -115/651, 0, -3/31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -34/93, -11/651, 0, 2/21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] gap> nonzero:= Filtered( [ 1 .. Length( sol ) ], i -> sol[i] <> 0 ); [ 1, 2, 3, 4, 6, 7, 9, 10, 12, 16, 25, 27, 106, 107, 109, 120 ] gap> sol:= sol{ nonzero };;
Now we transfer this linear combination to the character tables which are given in our situation.
Those constituents that are induced from subgroups of \(H\) above \(W\) can be identified uniquely via their degrees and their values distribution; we compute these characters in the character table of \(U/W\) obtained as a factor table of the character table of \(U/N\), lift them back to \(U/N\), induce them to \(G/N\), inflate them to \(G\), and then induce them fo \(M\).
gap> a8degrees:= List( above{ Filtered( nonzero, > x -> x <= Length( above ) ) }, > x -> x[1] ) / 31; [ 1, 8, 15, 28, 56, 56, 70, 105, 120, 168, 336, 336 ] gap> a8tbl:= subtbl / [ 1, 2 ];; gap> invtoa8:= InverseMap( GetFusionMap( subtbl, a8tbl ) );; gap> nsg:= ClassPositionsOfNormalSubgroups( k );; gap> nn:= First( nsg, x -> Sum( SizesConjugacyClasses( k ){ x } ) = 6*2^14 );; gap> a8tbl_other:= k / nn;; gap> g:= CharacterTable( "MC2B" ); CharacterTable( "2^1+24.Co1" ) gap> constit:= [];; gap> for i in [ 1 .. Length( a8degrees ) ] do > cand:= PermChars( a8tbl_other, rec( torso:= [ a8degrees[i] ] ) ); > filt:= Filtered( perms, x -> x^tbl = above[ nonzero[i] ] ); > filt:= List( filt, x -> CompositionMaps( x, invtoa8 ) ); > cand:= Filtered( cand, > x -> ForAny( filt, y -> Collected( x ) = Collected(y) ) ); > Add( constit, List( Induced( Restricted( Induced( > Restricted( cand, k ), co1 ), g ), m ), ValuesOfClassFunction ) ); > od; gap> List( constit, Length ); [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
Dealing with the remaining constituents is more involved. For a permutation character \(1_{{X/V}}^{{U/V}}\), we compute \(1_{{WX/V}}^{{U/V}}\), a character whose degree is half as large and which can be regarded as a character of \(U/W\). This character can be treated like the ones above: We lift it to \(U/N\), induce it to \(G/N\), and inflate it to \(G/Z(G)\); let this character be \(1_Y^{{G/Z(G)}}\), for some subgroup \(Y\). Then we compute the possible permutation characters of \(G/Z(G)\) that can be induced from a subgroup of index two inside \(Y\), inflate these characters to \(G\) and then induce them to \(M\).
gap> downdegrees:= List( index2{ Filtered( nonzero, > x -> x > Length( above ) ) > - Length( above ) }, > x -> x[1] ) / 31; [ 30, 210, 210, 1920 ] gap> f:= g / ClassPositionsOfCentre( g );; gap> forders:= OrdersClassRepresentatives( f );; gap> inv:= InverseMap( GetFusionMap( g, f ) );; gap> for j in [ 1 .. Length( downdegrees ) ] do > chars:= []; > cand:= PermChars( a8tbl_other, rec( torso:= [ downdegrees[j]/2 ] ) ); > filt:= Filtered( perms, x -> x^tbl = index2[ nonzero[ > j + Length( a8degrees ) ] - Length( above ) ] ); > filt:= Induced( subtbl, a8tbl, filt, > GetFusionMap( subtbl, a8tbl )); > cand:= Filtered( cand, x -> ForAny( filt, > y -> Collected( x ) = Collected( y ) ) ); > cand:= Restricted( Induced( Restricted( cand, k ), co1 ), g ); > for chi in cand do > cchi:= CompositionMaps( chi, inv ); > upper:= []; > pphi:= []; > for i in [ 1 .. NrConjugacyClasses( f ) ] do > if forders[i] mod 2 = 1 then > pphi[i]:= 2 * cchi[i]; > elif cchi[i] = 0 then > pphi[i]:= 0; > fi; > upper[i]:= 2* cchi[i]; > od; > Append( chars, PermChars( f, rec( torso:= ShallowCopy( pphi ), > upper:= upper, > normalsubgroup:= [ 1 .. 4 ], > nonfaithful:= cchi ) ) ); > od; > Add( constit, List( Induced( Restricted( chars, g ), m ), > ValuesOfClassFunction ) ); > od; gap> List( constit, Length ); [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 10, 10, 2 ]
Now we form the possible linear combinations.
gap> cand:= List( Cartesian( constit ), l -> sol * l );; gap> m:= CharacterTable( "M" ); CharacterTable( "M" ) gap> morders:= OrdersClassRepresentatives( m );; gap> for x in cand do > for i in [ 1 .. Length( morders ) ] do > if morders[i] mod 31 = 0 or morders[i] mod 21 = 0 then > Unbind( x[i] ); > fi; > od; > od;
Exactly one of the candidates has only integral values.
gap> cand:= Filtered( cand, x -> ForAll( x, IsInt ) ); [ [ 391965121389536908413379198941796875, 23914487292951376996875, 474163138042468875, 9500455925885925, 646346515815, 334363486275, 954161764875, 147339103275, 1481392395, 1313281515, 0, 8203125, 9827885925, 1216215, 91556325, 9388791, 115911, 587331, 874650, 0, 79515, 581955, 336375, 104371, 62331, 36855, 0, 0, 0, 0, 28125, 525, 1125, 0, 188325, 16767, 88965, 2403, 9477, 1155, 891, 207, 351, 627, 0, 0, 4410, 1498, 0, 0, 0, 30, 150, 91, 151, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 125, 0, 5, 5,,,,, 0, 0, 0, 0, 141, 45, 27, 61, 27, 9, 9, 7, 3, 15, 0, 0, 0, 0, 0, 98, 74, 42, 0, 0, 30, 0, 0, 0, 6, 6, 6,,, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0,,,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,,,, 0, 0, 0, 0, 0, 0,,, 0, 0, 0, 0, 0, 0,, 0, 0, 0 ] ]
Now we compute the possible permutation characters that have the prescribed values.
gap> cand:= PermChars( m, rec( torso:= cand[1] ) ); [ Character( CharacterTable( "M" ), [ 391965121389536908413379198941796875, 23914487292951376996875, 474163138042468875, 9500455925885925, 646346515815, 334363486275, 954161764875, 147339103275, 1481392395, 1313281515, 0, 8203125, 9827885925, 1216215, 91556325, 9388791, 115911, 587331, 874650, 0, 79515, 581955, 336375, 104371, 62331, 36855, 0, 0, 0, 0, 28125, 525, 1125, 0, 188325, 16767, 88965, 2403, 9477, 1155, 891, 207, 351, 627, 0, 0, 4410, 1498, 0, 0, 0, 30, 150, 91, 151, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 125, 0, 5, 5, 210, 0, 42, 0, 0, 0, 0, 0, 141, 45, 27, 61, 27, 9, 9, 7, 3, 15, 0, 0, 0, 0, 0, 98, 74, 42, 0, 0, 30, 0, 0, 0, 6, 6, 6, 3, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 18, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
There is only one candidate, so we have found the permutation character.
According to the Atlas of Finite Groups [CCN+85, p. 234], the Monster group \(M\) has a class of maximal subgroups \(H\) of the type \(2^{{10+16}}.O_{10}^+(2)\). Currently the character table of \(H\) and the class fusion into \(M\) are not available in GAP. We are interested in the permutation character \(1_H^M\), and we will compute it without this information.
The subgroup \(H\) normalizes an elementary abelian group \(E\) of order \(2^{10}\) which contains \(496\) involutions in the class 2A and \(527\) involutions in the class 2B. Let \(V\) denote the normal subgroup of order \(2^{26}\) in \(H\), and set \(\bar{H} = H/N\). Since the smallest two indices of maximal subgroups of \(\bar{H}\) are \(496\) and \(527\), respectively, \(H\) acts transitively on both the 2A and the 2B involutions in \(E\), and the centralizers of these involutions contain \(V\).
gap> Hbar:= CharacterTable( "O10+(2)" );; gap> U_Abar:= CharacterTable( "O10+(2)M1" ); CharacterTable( "S8(2)" ) gap> Index( Hbar, U_Abar ); 496 gap> U_Bbar:= CharacterTable( "O10+(2)M2" ); CharacterTable( "2^8:O8+(2)" ) gap> Index( Hbar, U_Bbar ); 527
We fix a 2A involution \(z_A\) in \(E\), and set \(G_A = C_M(z_A)\) and \(U_A = C_H(z_A)\). So \(G_A\) has the structure \(2.B\) and \(U_A\) has the structure \(2^{{10+16}}.S_8(2)\). From the list of maximal subgroups of \(B\) we see that the image of \(G_A\) under the natural epimorphism from \(G_A\) to \(B\) is a maximal subgroup of \(B\) and has the structure \(2^{{9+16}}.S_8(2)\).
gap> b:= CharacterTable( "B" ); CharacterTable( "B" ) gap> Horder:= 2^26 * Size( Hbar ); 1577011055923770163200 gap> order:= Horder / ( 2 * 496 ); 1589728887019929600 gap> maxes:= List( Maxes( b ), CharacterTable );; gap> filt:= Filtered( maxes, t -> Size( t ) mod order = 0 ); [ CharacterTable( "2^(9+16).S8(2)" ) ] gap> List( filt, t -> Size( t ) / order ); [ 1 ] gap> u1:= filt[1]; CharacterTable( "2^(9+16).S8(2)" )
Analogously, we fix a 2B involution \(z_B\) in \(E\), and set \(G_B = C_M(z_B)\) and \(U_B = C_H(z_B)\), Further, let \(N\) be the extraspecial normal subgroup of order \(2^{25}\) in \(G_B\). So \(G_B\) has the structure \(2^{{1+24}}_+.Co_1\), and \(U_B\) has index \(527\) in \(G_B\). From the list of maximal subgroups of \(Co_1\) we see that the image of \(U_B\) under the natural epimorphism from \(G_B\) to \(Co_1\) is a maximal subgroup of \(Co_1\) and has the structure \(2^{{1+8}}_+.O_8^+(2)\).
gap> co1:= CharacterTable( "Co1" );; gap> order:= Horder / ( 2^25 * 527 ); 89181388800 gap> maxes:= List( Maxes( co1 ), CharacterTable );; gap> filt:= Filtered( maxes, t -> Size( t ) mod order = 0 ); [ CharacterTable( "2^(1+8)+.O8+(2)" ) ] gap> List( filt, t -> Size( t ) / order ); [ 1 ] gap> u2:= filt[1]; CharacterTable( "2^(1+8)+.O8+(2)" )
First we compute the permutation characters \(\pi_A = 1_{{U_A}}^M\) and \(\pi_B = 1_{{U_B}}^M\).
gap> m:= CharacterTable( "M" ); CharacterTable( "M" ) gap> 2b:= CharacterTable( "MC2A" ); CharacterTable( "2.B" ) gap> mm:= CharacterTable( "MC2B" ); CharacterTable( "2^1+24.Co1" ) gap> pi_A:= RestrictedClassFunction( TrivialCharacter( u1 )^b, 2b )^m;; gap> pi_B:= RestrictedClassFunction( TrivialCharacter( u2 )^co1, mm )^m;;
The degree of \(1_H^M\) is of course known.
gap> torso:= [ Size( m ) / Horder ]; [ 512372707698741056749515292734375 ]
Next we compute some zero values of \(1_H^M\), using the following conditions.
For \(g \in M\), if \(|g|\) does not divide \(|H|\) or if \(|g|\) is not the product of an element order in \(H/V\) and a \(2\)-power. (In fact we could use that the exponent of \(V\) is \(4\), but this would not improve the result.)
Let \(U \leq H \leq G\), and let \(p\) be a prime that does not divide \([H:U]\). Then \(U\) contains a Sylow \(p\) subgroup of \(H\), so each element of order \(p\) in \(H\) is conjugate in \(H\) to an element in \(U\). For \(g \in G\), \(g = g_p h\), where the order of \(g_p\) is a power of \(p\) such that \(1_U^G(g_p) = 0\) holds, we have \(1_H^G(g) = 0\). We apply this to \(U \in \{ U_A, U_B \}\).
gap> morders:= OrdersClassRepresentatives( m );; gap> 2parts:= Union( [ 1 ], Filtered( Set( morders ), > x -> IsPrimePowerInt( x ) and IsEvenInt( x ) ) ); [ 1, 2, 4, 8, 16, 32 ] gap> factorders:= Set( OrdersClassRepresentatives( Hbar ) );; gap> primes_A:= Filtered( PrimeDivisors( Horder ), p -> 496 mod p <> 0 ); [ 3, 5, 7, 17 ] gap> primes_B:= Filtered( PrimeDivisors( Horder ), p -> 527 mod p <> 0 ); [ 2, 3, 5, 7 ] gap> primes:= Union( primes_A, primes_B );; gap> n:= NrConjugacyClasses( m );; gap> for i in [ 1 .. n ] do > if Horder mod morders[i] <> 0 then > torso[i]:= 0; > elif ForAll( factorders, x -> not morders[i] / x in 2parts ) then > torso[i]:= 0; > else > for p in primes do > if morders[i] mod p = 0 then > pprime:= morders[i]; > while pprime mod p = 0 do pprime:= pprime / p; od; > pos:= PowerMap( m, pprime )[i]; > if p in primes_A and pi_A[ pos ] = 0 then > torso[i]:= 0; > elif p in primes_B and pi_B[ pos ] = 0 then > torso[i]:= 0; > fi; > fi; > od; > fi; > od; gap> torso; [ 512372707698741056749515292734375,,,,, 0,,,,,,,,,,,, 0,, 0,,,,,,,,,, ,,,, 0,,,, 0,,,,,, 0, 0, 0,,, 0,,,, 0,,,,,,,,,, 0,,,,,,,, 0, 0, 0, 0, 0, 0, 0,,,,, 0,,,,, 0, 0, 0, 0, 0, 0,,,, 0, 0,,,,, 0,,,,,,, 0, 0, , 0, 0,,,,, 0, 0, 0, 0, 0,,,,, 0,, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, , 0,, 0, 0, 0, 0,, 0, 0, 0, 0, 0,,,,,, 0,,, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
Now we want to compute as many nonzero values of \(1_H^M\) as possible, using the same approach as in the previous sections. For that, we first compute several permutation characters \(1_X^M\), for subgroups \(X\) with the property \(V < X < U_A\) or \(V < X < U_B\). Then we find several subsets \(C\) of \(M\), each being a union of conjugacy classes of \(M\) such that \((1_H)_{{C \cap H}}\) is a linear combination of the characters \(1_X^H\), restricted to \(C \cap H\). This yields the values of \(1_H^M\) on the classes in \(C\).
The actual computations are performed with the characters \(1_{{X/V}}^{{H/V}}\). So we build two parallel lists cand and candbar of permutation characters of \(M\) and of \(H/V\), respectively. For that, we write two small GAP functions:
In the function AddSubgroupOfS82, we choose a subgroup \(Y\) of \(S_8(2) \cong U_A/V\), compute \(1_Y^{{U_A/V}}\), inflate it to a character of \(U_A\), induce this character to \(B\), inflate the result to \(G_A\), and finally induce this character to \(M\).
In the function AddSubgroupOfO8p2, we choose a subgroup \(Y\) of the factor group \(F \cong O_8^+(2)\) of \(U_B/N\), compute \(1_Y^F\), inflate it to a character of \(U_B/N\), induce this to a character of \(G_B/N \cong Co_1\), inflate this to a character of \(G_B\), and finally induce this character to \(M\).
One difficulty in this case is that choosing a subgroup \(X/V\) of \(H/V\) involves fixing the class fusion into \(H/V\), but it is not clear which is a compatible class fusion of the corresponding subgroup \(X\) into \(M\); therefore, each entry of cand is in fact not the permutation character of \(M\) in question but a list of possibilities.
gap> cand:= [ [ pi_A ], [ pi_B ] ];; gap> candbar:= [ TrivialCharacter( U_Abar )^Hbar, > TrivialCharacter( U_Bbar )^Hbar ];; gap> AddSubgroupOfS82:= function( subname ) > local psis82; > > psis82:= TrivialCharacter( CharacterTable( subname ) )^U_Abar; > Add( cand, [ Restricted( Restricted( psis82, u1 )^b, 2b )^m ] ); > Add( candbar, psis82 ^ Hbar ); > end;; gap> tt1:= CharacterTable( "O8+(2)" ); CharacterTable( "O8+(2)" ) gap> AddSubgroupOfO8p2:= function( subname ) > local psi, list, char; > > psi:= TrivialCharacter( CharacterTable( subname ) )^tt1; > list:= []; > for char in Orbit( AutomorphismsOfTable( tt1 ), psi, Permuted ) do > AddSet( list, Restricted( Restricted( char, u2 ) ^ co1, mm ) ^ m ); > od; > Add( cand, list ); > Add( candbar, Restricted( psi, U_Bbar ) ^ Hbar ); > end;;
Now we choose the subgroups that will turn out to be sufficient for our computations.
gap> AddSubgroupOfS82( "O8+(2).2" ); gap> AddSubgroupOfO8p2( "S6(2)" ); gap> AddSubgroupOfS82( "O8-(2).2" ); gap> AddSubgroupOfS82( "A10.2" ); gap> AddSubgroupOfS82( "S4(4).2" ); gap> AddSubgroupOfS82( "L2(17)" ); gap> AddSubgroupOfO8p2( "A9" ); gap> AddSubgroupOfO8p2( "2^6:A8" ); gap> AddSubgroupOfO8p2( "(3xU4(2)):2" ); gap> AddSubgroupOfO8p2( "(A5xA5):2^2" ); gap> AddSubgroupOfS82( "S8(2)M4" );
In the case of \(A_5 < S_8(2)\), the function AddSubgroupOfS82 does not work because there are several class fusions of \(A_5\) into \(S_8(2)\). We choose one fusion; the fact that it really describes an embedding of an \(A_5\) type subgroup of \(S_8(2)\) can be checked using the function NrPolyhedralSubgroups (Reference: NrPolyhedralSubgroups).
gap> a5:= CharacterTable( "A5" );; gap> fus:= PossibleClassFusions( a5, U_Abar )[1];; gap> NrPolyhedralSubgroups( U_Abar, fus[2], fus[3], fus[4] ); rec( number := 548352, type := "A5" ) gap> psis82:= Induced( a5, U_Abar, [ TrivialCharacter( a5 ) ], fus )[1];; gap> Add( cand, [ Restricted( Restricted( psis82, u1 )^b, 2b )^m ] ); gap> Add( candbar, psis82 ^ Hbar ); gap> List( cand, Length ); [ 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1 ]
The following function takes a condition on conjugacy classes in terms of their element orders, which gives a set \(C\) of elements in \(M\). It forms the corresponding set of elements in \(H/V\) and tries to express the restriction of \(1_{{H/V}}\) as a linear combination of the characters \(1_X^{{H/V}}\) that are stored in the list candbar. If this works and if the corresponding linear combination of the candidates in cand is unique, the newly found values of \(1_H^M\) are entered into the list torso.
gap> Hbarorders:= OrdersClassRepresentatives( Hbar );; gap> TryCondition:= function( cond ) > local pos, sol, lincomb, oldknown, i; > > pos:= Filtered( [ 1 .. Length( Hbarorders ) ], > i -> cond( Hbarorders[i] ) ); > sol:= SolutionMat( candbar{[1..Length(candbar)]}{ pos }, > ListWithIdenticalEntries( Length( pos ), 1 ) ); > if sol = fail then > return "no solution"; > fi; > > pos:= Filtered( [ 1 .. Length( morders) ], i -> cond( morders[i] ) ); > lincomb:= Filtered( Set( Cartesian( cand ), x -> sol * x ), > x -> ForAll( pos, i -> IsPosInt( x[i] ) or x[i] = 0 ) ); > if Length( lincomb ) <> 1 then > return "solution is not unique"; > fi; > > lincomb:= lincomb[1];; > oldknown:= Number( torso ); > for i in pos do > if IsBound( torso[i] ) then > if torso[i] <> lincomb[i] then > Error( "contradiction of new and known value at position ", i ); > fi; > elif not IsInt( lincomb[i] ) or lincomb[i] < 0 then > Error( "new value at position ", i, " is not a nonneg. integer" ); > fi; > torso[i]:= lincomb[i]; > od; > return Concatenation( "now ", String( Number( torso ) ), " values (", > String( Number( torso ) - oldknown ), " new)" ); > end;;
This procedure makes sense only if the elements of \(H\) that satisfy the condition are contained in the full preimage of the classes of \(H/V\) that satisfy the condition. Note that this is in fact the case for the conditions used below. This is clear for condition concerning only odd element orders, because \(V\) is a \(2\)-group. Also the set of all elements of the orders \(9\), \(18\), and \(36\) is such a "closed" set, since \(M\) has no elements of order \(72\). Finally, the set of all elements of the orders \(1\), \(2\), and \(4\) in \(H\) is admissible because it is contained in the preimage of the set of all elements of these orders in \(H/V\).
gap> TryCondition( x -> x mod 7 = 0 and x mod 3 <> 0 ); "now 99 values (7 new)" gap> TryCondition( x -> x mod 17 = 0 and x mod 3 <> 0 ); "now 102 values (3 new)" gap> TryCondition( x -> x mod 5 = 0 and x mod 3 <> 0 ); "now 119 values (17 new)" gap> TryCondition( x -> 4 mod x = 0 ); "now 125 values (6 new)" gap> TryCondition( x -> 9 mod x = 0 ); "now 129 values (4 new)" gap> TryCondition( x -> x in [ 9, 18, 36 ] ); "now 138 values (9 new)"
Possible constituents of \(1_H^M\) are those rational irreducible characters of \(M\) that are constituents of \(\pi^M\).
gap> constit:= Filtered( RationalizedMat( Irr( m ) ), > x -> ScalarProduct( m, x, pi_A ) <> 0 > and ScalarProduct( m, x, pi_B ) <> 0 );;
For the missing values, we can provide at least lower bounds, using that \(U \leq H \leq G\) implies \(1_H^G(g) \geq 1_U^G(g) / [H:U] = [G:H] \cdot 1_U^G(g) / 1_U^G(1)\).
gap> lower:= [];; gap> Hindex:= Size( m ) / Horder; 512372707698741056749515292734375 gap> for i in [ 1 .. NrConjugacyClasses( m ) ] do > lower[i]:= Maximum( pi_A[i] / ( pi_A[1] / Hindex ), > pi_B[i] / ( pi_B[1] / Hindex ) ); > if not IsInt( lower[i] ) then > lower[i]:= Int( lower[i] + 1 ); > fi; > od;
Now we compute the possible permutation characters that have the prescribed values, are compatible with the given lower bounds for values, and have only constituents in the given list.
gap> cand:= PermChars( m, rec( torso:= torso, chars:= constit, > lower:= lower, normalsubgroup:= [ 1 .. NrConjugacyClasses( m ) ], > nonfaithful:= TrivialCharacter( m ) ) ); [ Character( CharacterTable( "M" ), [ 512372707698741056749515292734375, 405589064025344574375, 29628786742129575, 658201521662685, 215448838605, 0, 121971774375, 28098354375, 335229607, 108472455, 164587500, 4921875, 2487507165, 2567565, 26157789, 6593805, 398925, 0, 437325, 0, 44983, 234399, 90675, 21391, 41111, 12915, 6561, 6561, 177100, 7660, 6875, 315, 275, 0, 113373, 17901, 57213, 0, 4957, 1197, 909, 301, 397, 0, 0, 0, 3885, 525, 0, 2835, 90, 45, 0, 103, 67, 43, 28, 81, 189, 9, 9, 9, 0, 540, 300, 175, 20, 15, 7, 420, 0, 0, 0, 0, 0, 0, 0, 165, 61, 37, 37, 0, 9, 9, 13, 5, 0, 0, 0, 0, 0, 0, 77, 45, 13, 0, 0, 45, 115, 19, 10, 0, 5, 5, 9, 9, 1, 1, 0, 0, 4, 0, 0, 9, 9, 3, 1, 0, 0, 0, 0, 0, 0, 4, 1, 1, 0, 24, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 3, 1, 1, 2, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ]
There is only one candidate, so we have found the permutation character.
We compute the character of the Baby Monster that is induced from the trivial character of a Sylow \(2\)-subgroup. (Gabriel Navarro had asked me how GAP can compute this character.) We start with the computation of those transitive permutation characters of the symmetric group on five points that have degree \(15\). Note that the function PermChars (Reference: PermChars) computes in general only candidates, but here we are sure that the result consists of permutation characters because it is unique.
gap> t:= CharacterTable( "S5" ); CharacterTable( "A5.2" ) gap> pi:= PermChars( t, rec( torso:= [ 15 ] ) ); [ Character( CharacterTable( "A5.2" ), [ 15, 3, 0, 0, 3, 1, 0 ] ) ]
Next, we regard this character as a character of the group \(2^5:S_5\) that occurs as a maximal subgroup of index \(231\) in \(M_{22}:2\).
gap> m222:= CharacterTable( "M22.2" ); CharacterTable( "M22.2" ) gap> mx:= List( Maxes( m222 ), CharacterTable );; gap> mx:= Filtered( mx, x -> Size( m222 ) / Size( x ) = 231 ); [ CharacterTable( "M22.2M4" ) ] gap> pi:= pi[1]{ GetFusionMap( mx[1], t ) }; [ 15, 15, 3, 3, 3, 0, 0, 3, 3, 1, 1, 0, 15, 15, 3, 3, 3, 0, 0, 3, 3, 1, 1, 0 ]
We induce this character to \(M_{22}:2\). (Note that this is the character that is induced from the trivial character of a Sylow \(2\)-subgroup of \(M_{22}:2\).)
gap> pi:= InducedClassFunction( mx[1], pi, m222 ); ClassFunction( CharacterTable( "M22.2" ), [ 3465, 105, 0, 9, 5, 0, 0, 0, 0, 1, 0, 189, 45, 9, 13, 0, 1, 0, 0, 0, 0 ] )
Next, we regard this character as a character of the group \(2^{10}:M_{22}:2\) that occurs as a maximal subgroup of index \(46575\) in \(Co_2\).
gap> co2:= CharacterTable( "Co2" ); CharacterTable( "Co2" ) gap> mx:= List( Maxes( co2 ), CharacterTable );; gap> mx:= Filtered( mx, x -> Size( co2 ) / Size( x ) = 46575 ); [ CharacterTable( "2^10:m22:2" ) ] gap> pi:= pi{ GetFusionMap( mx[1], m222 ) }; [ 3465, 3465, 3465, 3465, 105, 105, 105, 105, 105, 105, 105, 105, 0, 0, 0, 0, 0, 9, 9, 9, 9, 9, 9, 5, 5, 5, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 189, 189, 189, 189, 189, 189, 45, 45, 45, 45, 9, 9, 9, 9, 13, 13, 13, 13, 13, 13, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
We induce this character to \(Co_2\).
gap> pi:= InducedClassFunction( mx[1], pi, co2 ); ClassFunction( CharacterTable( "Co2" ), [ 161382375, 626535, 162855, 27495, 0, 0, 6615, 3975, 2727, 855, 567, 975, 115, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, 51, 19, 27, 35, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] )
Next, we regard this character as a character of the group \(2^{{1+22}}.Co_2\) that occurs as a maximal subgroup of index \(11707448673375\) in the Baby Monster.
gap> b:= CharacterTable( "B" ); CharacterTable( "B" ) gap> mx:= List( Maxes( b ), CharacterTable );; gap> mx:= Filtered( mx, x -> Size( b ) / Size( x ) = 11707448673375 ); [ CharacterTable( "2^(1+22).Co2" ) ] gap> pi:= pi{ GetFusionMap( mx[1], co2 ) };; gap> pi[1]; 161382375
We induce this character to the Baby Monster.
gap> pi:= InducedClassFunction( mx[1], pi, b ); ClassFunction( CharacterTable( "B" ), [ 1889375872099856765625, 2609385408855225, 62316674429625, 207818526825, 268788490425, 0, 0, 13052741625, 7537207545, 128298681, 270580905, 46366425, 74315385, 35633385, 3937689, 201825, 1233225, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 713097, 241425, 320625, 88521, 275265, 57705, 19305, 20089, 9441, 6489, 2577, 1825, 5345, 753, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 273, 417, 105, 97, 185, 33, 9, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] )
We compute the character of the double cover \(2.B\) of the Baby Monster that is induced from the trivial character of a subgroup \(U\) of the structure \(2^{1+22}.McL\).
This subgroup occurs as the intersection of two conjugates of \(2.B\) inside the Monster group \(M\). More precisely, we consider \(2.B\) as the centralizer of an involution \(a\) in \(M\), and we are interested in the permutation action of \(M\) on the cosets of \(2.B\) (or, equivalently, on the conjugacy class in \(M\) of this involution). The restriction of this action to \(2.B\) has nine orbits. One of them has point stabilizer \(U\).
Background information can be found in [GJMS89]. The decomposition into the nine orbits appears in Definition (3.4.9) on p 587, and our orbit is characterized in Table VII (on p. 582) by the facts that its points \(c\) have order \(4\) and the squares of \(a c\) lie in the class 2B of \(M\). This implies that \(a\) and \(c\) do not commute, hence \(a\) does not lie in \(U\).
From this description, we know that \(U\) is a subgroup of a maximal subgroup of the type \(2^{2+22}.Co_2\) in \(2.B\), and the group \(\langle U, a \rangle\) has the type \(2^{2+22}.McL\).
Thus we can proceed in two steps. First we induce the trivial character of \(\langle U, a \rangle\) to \(2.B\). Then we use the variant of the GAP function PermChars (Reference: PermChars) that allows us to prescribe the permutation character of the closure with a normal subgroup, which is \(\langle a \rangle\) in our case.
The first step can be performed by inducing the trivial character of \(McL\) to \(Co_2\), \(\ldots\)
gap> mcl:= CharacterTable( "McL" ); CharacterTable( "McL" ) gap> co2:= CharacterTable( "Co2" ); CharacterTable( "Co2" ) gap> ind:= Induced( mcl, co2, [ TrivialCharacter( mcl ) ] )[1]; Character( CharacterTable( "Co2" ), [ 47104, 0, 1024, 0, 16, 160, 0, 0, 0, 0, 64, 0, 0, 4, 24, 16, 0, 0, 0, 16, 0, 8, 0, 0, 0, 0, 0, 8, 4, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ] )
\(\ldots\) regarding this character as a character of \(2^{1+22}.Co_2\), \(\ldots\)
gap> m:= CharacterTable( "BM2" ); CharacterTable( "2^(1+22).Co2" ) gap> infl:= ind{ GetFusionMap( m, co2 ) }; [ 47104, 47104, 47104, 47104, 47104, 47104, 47104, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 16, 16, 16, 160, 160, 160, 160, 160, 160, 160, 160, 160, 160, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 4, 24, 24, 24, 24, 24, 24, 24, 24, 16, 16, 16, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 8, 8, 8, 8, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1 ]
\(\ldots\) inducing this character to \(B\), \(\ldots\)
gap> b:= CharacterTable( "B" ); CharacterTable( "B" ) gap> ind:= Induced( m, b, [ infl ] )[1]; ClassFunction( CharacterTable( "B" ), [ 551467662310656000, 186911262720, 272993634304, 0, 634521600, 194594400, 69984, 8495104, 17465344, 129024, 276480, 2073600, 16384, 798720, 46080, 0, 5120, 138600, 1000, 110880, 252000, 112480, 432, 12960, 0, 1312, 8352, 864, 432, 0, 2520, 0, 2880, 2880, 3072, 2880, 0, 0, 256, 64, 1152, 576, 640, 192, 96, 0, 108, 2520, 744, 0, 104, 120, 40, 30, 160, 16, 1120, 1024, 0, 0, 96, 288, 64, 144, 0, 96, 0, 108, 16, 48, 0, 32, 12, 0, 0, 0, 168, 0, 104, 48, 0, 4, 0, 0, 0, 0, 32, 16, 8, 8, 0, 24, 12, 4, 0, 0, 0, 0, 24, 4, 24, 24, 0, 0, 0, 0, 4, 0, 0, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 8, 0, 16, 8, 4, 0, 0, 0, 0, 0, 4, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] )
\(\ldots\) and regarding the result as a character of \(2.B\).
gap> 2b:= CharacterTable( "2.B" ); CharacterTable( "2.B" ) gap> infl:= ind{ GetFusionMap( 2b, b ) }; [ 551467662310656000, 551467662310656000, 186911262720, 272993634304, 272993634304, 0, 634521600, 194594400, 194594400, 69984, 69984, 8495104, 17465344, 129024, 276480, 2073600, 2073600, 16384, 798720, 46080, 0, 5120, 138600, 138600, 1000, 1000, 110880, 252000, 112480, 112480, 432, 12960, 0, 1312, 1312, 8352, 864, 864, 432, 0, 2520, 2520, 0, 2880, 2880, 3072, 2880, 0, 0, 256, 64, 1152, 576, 576, 640, 192, 96, 0, 0, 108, 108, 2520, 744, 744, 0, 104, 104, 120, 40, 40, 30, 30, 160, 16, 1120, 1024, 0, 0, 0, 96, 288, 64, 144, 144, 0, 96, 0, 108, 108, 16, 48, 0, 32, 12, 12, 0, 0, 0, 0, 168, 0, 104, 104, 48, 0, 0, 4, 4, 0, 0, 0, 0, 32, 16, 8, 8, 8, 0, 0, 24, 12, 4, 4, 0, 0, 0, 0, 0, 0, 24, 4, 24, 24, 0, 0, 0, 0, 0, 4, 0, 0, 0, 6, 6, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 0, 16, 8, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
Now we have the character \(\psi\) that represents the "nonfaithful half" of the desired permutation character. We have to "fill it up" with faithful characters of \(2.B\) of total degree \(\psi(1)\) such that the sum with \(\psi\) can be a permutation character of \(2.B\).
The GAP function PermChars (Reference: PermChars) is designed for this situation. We specify the normal subgroup \(N = \langle a \rangle\) by listing the positions of its conjugacy classes in the character table of \(2.B\), we enter the known permutation character \(1_{{UN}}^{{2.B}}\), and of course we specify the degree of the possible permutation characters.
gap> centre:= ClassPositionsOfCentre( 2b ); [ 1, 2 ] gap> pi:= PermChars( 2b, rec( torso:= [ 2 * infl[1], 0 ], > normalsubgroup:= centre, > nonfaithful:= infl ) ); [ Character( CharacterTable( "2.B" ), [ 1102935324621312000, 0, 186911262720, 541790208000, 4197060608, 0, 634521600, 389188800, 0, 139968, 0, 8495104, 17465344, 129024, 276480, 4026240, 120960, 16384, 798720, 46080, 0, 5120, 277200, 0, 2000, 0, 110880, 252000, 190080, 34880, 432, 12960, 0, 2592, 32, 8352, 1728, 0, 432, 0, 5040, 0, 0, 2880, 2880, 3072, 2880, 0, 0, 256, 64, 1152, 1008, 144, 640, 192, 96, 0, 0, 216, 0, 2520, 960, 528, 0, 200, 8, 120, 80, 0, 60, 0, 160, 16, 1120, 1024, 0, 0, 0, 96, 288, 64, 216, 72, 0, 96, 0, 216, 0, 16, 48, 0, 32, 24, 0, 0, 0, 0, 0, 168, 0, 160, 48, 48, 0, 0, 8, 0, 0, 0, 0, 0, 32, 16, 8, 12, 4, 0, 0, 24, 12, 0, 8, 0, 0, 0, 0, 0, 0, 24, 4, 24, 24, 0, 0, 0, 0, 0, 4, 0, 0, 0, 6, 6, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 0, 16, 8, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ] gap> MatScalarProducts( 2b, Irr( 2b ), pi ); [ [ 1, 1, 2, 1, 2, 0, 2, 3, 2, 0, 0, 1, 4, 1, 2, 0, 3, 2, 0, 2, 0, 0, 2, 2, 0, 0, 2, 3, 1, 5, 0, 4, 3, 2, 0, 0, 3, 2, 0, 6, 4, 0, 1, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 5, 0, 5, 2, 0, 0, 2, 0, 0, 4, 1, 0, 2, 0, 4, 2, 4, 4, 3, 0, 2, 4, 2, 4, 0, 3, 0, 3, 2, 5, 0, 1, 0, 3, 1, 0, 1, 1, 2, 5, 3, 1, 1, 4, 5, 1, 1, 0, 3, 0, 0, 3, 2, 1, 1, 2, 1, 1, 4, 0, 3, 2, 3, 1, 3, 0, 1, 3, 0, 2, 2, 1, 3, 3, 0, 0, 2, 0, 0, 0, 0, 3, 0, 3, 3, 3, 1, 0, 3, 0, 4, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 3, 3, 0, 0, 0, 1, 1, 1, 1, 2, 3, 2, 0, 0, 2, 2, 4, 3, 5, 2, 4, 0, 0, 0, 0, 5, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 7, 0, 0, 1, 7, 7, 0, 0, 0, 1, 6, 4, 5, 0, 0, 3, 0, 0, 0, 0, 0, 4, 1, 1, 3, 8, 3, 2, 2, 5, 0, 1 ] ]
We are lucky: There is a unique solution, and its computation is quite fast.
This section shows the computations that are needed in order to show the following statements from [BG].
Proposition 2.2: Let \(S\) be a sporadic simple group and let \(P\) be a Sylow \(2\)-subgroup of \(S\). If \(1 \neq x \in S\), then \(S = \langle P, x^g \rangle\) for some \(g \in S\).
Theorem 7.1: Let \(S\) be a sporadic simple group and let \(p \leq q\) be primes each dividing \(|S|\). Then \(S\) can be generated by a Sylow \(p\)-subgroup and a Sylow \(q\)-subgroup.
A stronger version of Theorem 7.1: Let \(S\) be a sporadic simple group \(p\) be a prime dividing \(|S|\), and \(P\) be a Sylow \(p\)-subgroup of \(G\). If \(1 \neq x \in S\), then \(S = \langle P, x^g \rangle\) for some \(g \in S\).
First we show [BG, Proposition 2.2]. Let \(S\) be a sporadic simple group, fix a Sylow \(2\)-subgroup \(P\) of \(S\), and let \(x\) be a nonidentity element in \(S\). We use known information about maximal subgroups of \(S\) to show that \(x^S\) is not a subset of the union of those maximal subgroups in \(S\) that contain \(P\).
Let \(M\) be a maximal subgroup of \(S\) with the property \(P \leq M\). The number of \(S\)-conjugates of \(M\) that contain \(P\) is equal to \(|N_S(P)|/|N_M(P)| \leq [N_S(P):P]\), thus these subgroups can contain at most \([N_S(P):P] |x^S \cap M|\) elements from the class \(x^S\).
Thus the number of elements in \(x^S\) that generate a proper subgroup of \(S\) together with \(P\) is bounded from above by \([N_S(P):P] \sum_M |x^S \cap M|\), where the sum is taken over representatives \(M\) of conjugacy classes of maximal subgroups of odd index in \(S\).
Let \(1_M^S\) denote the permutation character of the action of \(S\) on the cosets of \(M\). We have \(|x^S \cap M| = |x^S| 1_M^S(x) / 1_M^S(1)\). Hence we are done when we show that
\[ [N_S(P):P] \sum_M 1_M^S(x) / 1_M^S(1) < 1 \]
holds.
The numbers \([N_S(P):P]\) can be read off from [Wil98, Table I]. Here we use the fact that the character tables of the Sylow \(2\)-normalizer of \(S\) is available except if \(S\) is one of \(Co_1\), \(J_4\), \(F_{3+}\), \(B\), or \(M\), and that the Sylow \(2\)-subgroup if self-normalizing in these cases.
gap> names:= AllCharacterTableNames( IsSporadicSimple, true, > IsDuplicateTable, false : OrderedBy:= Size ); [ "M11", "M12", "J1", "M22", "J2", "M23", "HS", "J3", "M24", "McL", "He", "Ru", "Suz", "ON", "Co3", "Co2", "Fi22", "HN", "Ly", "Th", "Fi23", "Co1", "J4", "F3+", "B", "M" ] gap> normindices:= rec( Co1:= 1, J4:= 1, F3\+:= 1, B:= 1, M:= 1 );; gap> for name in names do > n:= CharacterTable( Concatenation( name, "N2" ) ); > if n = fail then > Print( name, "\n" ); > else > 2part:= 2^Length( Positions( Factors( Size( n ) ), 2 ) ); > normindices.( name ):= Size( n ) / 2part; > fi; > od; Co1 J4 F3+ B M
For all sporadic simple groups \(S\) except the Monster group, the primitive permutation characters \(1_M^S\) can be computed from the data about maximal subgroups contained in GAP's library of character tables.
gap> maxbound:= [];; gap> for name in Filtered( names, x -> x <> "M" ) do > t:= CharacterTable( name ); > mx:= List( Maxes( t ), CharacterTable ); > odd:= Filtered( mx, s -> ( Size( t ) / Size( s ) ) mod 2 <> 0 ); > primperm:= List( odd, s -> TrivialCharacter( s )^t ); > sum:= normindices.( name ) * Sum( primperm, pi -> pi / pi[1] ); > Add( maxbound, > [ name, Maximum( sum{ [ 2 .. Length( sum ) ] } ) ] ); > od; gap> SortBy( maxbound, x -> - x[2] ); gap> maxbound[1]; [ "J2", 3/5 ]
We see that the left hand side of the above inequality is always less than or equal to \(3/5\), in particular it is less than \(1\).
The Monster group is known to contain exactly five classes of maximal subgroups of odd index, of the structures \(2^{1+24}.Co_1\) (the normalizer of a 2B element in the Monster), \(2^{10+16}.O_{10}^+(2)\), \(2^{2+11+22}.(M_{24} \times S_3)\), \(2^{5+10+20}.(S_3 \times L_5(2))\), \([2^{39}].(L_3(2) \times 3S_6)\). The corresponding permutation characters are known, see Section 8.16. First we read the information about the known primitive permutation characters of the Monster into the GAP session, and extract the primitive permutation characters of odd degree.
gap> dir:= DirectoriesPackageLibrary( "ctbllib", "data" );; gap> filename:= Filename( dir, "prim_perm_M.json" );; gap> Monster_prim_data:= EvalString( StringFile( filename ) )[2];; gap> Length( Monster_prim_data ); 44 gap> t:= CharacterTable( "M" );; gap> monstermaxindices:= [];; gap> monstermaxtables:= [];; gap> for entry in Monster_prim_data do > if Length( entry ) = 1 then > s:= CharacterTable( entry[1] ); > Add( monstermaxtables, s ); > Add( monstermaxindices, Size( t ) / Size( s ) ); > else > Add( monstermaxtables, fail ); > Add( monstermaxindices, entry[2][1] ); > fi; > od; gap> odd_prim:= [];; gap> for i in [ 1 .. Length( Monster_prim_data ) ] do > if monstermaxindices[i] mod 2 <> 0 then > if monstermaxtables[i] <> fail then > Add( odd_prim, TrivialCharacter( monstermaxtables[i] )^t ); > else > Add( odd_prim, Monster_prim_data[i][2] ); > fi; > fi; > od; gap> Length( odd_prim ); 5
Now we can use the same approach as before.
gap> sum:= normindices.M * Sum( odd_prim, pi -> pi / pi[1] );; gap> max:= Maximum( sum{ [ 2 .. Length( sum ) ] } ); 12784979/103007903752128375 gap> max < 10^-9; true
Next we show [BG, Theorem 7.1] and its stronger version stated above. Let us first assume that \(S\) is not the Monster.
As a first step, we generalize the approach from the above computations in order to check for which prime divisors \(p\) of \(|S|\) and for which nontrivial conjugacy classes \(x^S\) of \(S\) the group \(S\) is generated by a Sylow \(p\)-subgroup \(P\) together with a conjugate of \(x\).
The upper bound \([N_S(P):P]\) for \(|N_S(P)|/|N_M(P)|\), for a maximal subgroup \(M\) of \(S\) that contains \(P\), is not good enough in some of the cases considered here. Instead of it, we compute the upper bound \(u(S, M, p)\) which is defined as follows; we assume that we know \(|N_S(P)|\).
If \(P\) is cyclic then we can compute \(|N_M(P)|\) from the character table of \(M\), and set \(u(S, M, p) = |N_S(P)| / |N_M(P)|\).
Otherwise, if \(P\) is normal in \(M\), we set \(u(S, M, p) = |N_S(P)| / |M|\).
Otherwise, if we know a subgroup \(U\) of \(M\) such that \(P\) is a proper normal subgroup of \(U\), we set \(u(S, M, p) = |N_S(P)| / |U|\).
Otherwise, we set \(u(S, M, p) = |N_S(P)| / |P|\).
gap> upper_bound:= function( tblS, tblM, p ) > local ppart, ppartposS, ppartposM, n, N_S, f, subname, u; > > ppart:= Product( Filtered( Factors( Size( tblS ) ), x -> x = p ), 1 ); > ppartposS:= Positions( OrdersClassRepresentatives( tblS ), ppart ); > if 0 < Length( ppartposS ) then > # P is cyclic. > if tblM = fail then > return ( SizesCentralizers( tblS )[ ppartposS[1] ] * Phi( ppart ) > / Length( ppartposS ) ) / ppart; > else > ppartposM:= Positions( OrdersClassRepresentatives( tblM ), ppart ); > return ( SizesCentralizers( tblS )[ ppartposS[1] ] * Phi( ppart ) > / Length( ppartposS ) ) / > ( SizesCentralizers( tblM )[ ppartposM[1] ] * Phi( ppart ) > / Length( ppartposM ) ); > fi; > fi; > > # Compute |N_S(P)|. > n:= CharacterTable( Concatenation( Identifier( tblS ), "N", > String( p ) ) ); > if n <> fail then > N_S:= Size( n ); > elif p = 2 then > N_S:= ppart * normindices.( Identifier( tblS ) ); > elif Identifier( tblS ) = "M" and p = 3 then > # The Sylow 3-normalizer is contained in 3^(3+2+6+6):(L3(3)xSD16) > N_S:= ppart * 2^6; > elif Identifier( tblS ) = "F3+" and p = 3 then > N_S:= ppart * 8; > else > Error( "cannot compute |N_S(P)|" ); > fi; > > if tblM = fail then > return N_S / ppart; > elif Sum( SizesConjugacyClasses( tblM ){ > ClassPositionsOfPCore( tblM, p ) } ) = ppart then > # P is normal in M. > return N_S / Size( tblM ); > fi; > > # Inspect known character tables of subgroups of M. > f:= N_S / ppart; > for subname in NamesOfFusionSources( tblM ) do > u:= CharacterTable( subname ); > if ClassPositionsOfKernel( GetFusionMap( u, tblM ) ) = [ 1 ] and > Sum( SizesConjugacyClasses( u ){ > ClassPositionsOfPCore( u, p ) } ) = ppart then > f:= Minimum( f, N_S / Size( u ) ); > fi; > od; > > return f; > end;;
We run over the sporadic simple groups (except the Monster), and collect in the list badcases_strong those "bad" prime divisors \(p\) of \(|S|\) and conjugacy class representatives \(x\) of nonidentity elements in \(S\) for which
\[ \sum_M u(S, M, p) 1_M^S(x) / 1_M^S(1) \geq 1 \]
holds, where the sum is taken over representatives \(M\) of conjugacy classes of maximal subgroups of \(S\) whose index in \(S\) is coprime to \(p\). In these cases, we have to find other arguments.
For the proof of [BG, Theorem 7.1], we can discard all those entries from the list of "bad" \(p\) and \(x\) where \(x\) is not a \(q\)-element, for some prime \(q\), or where another nonidentity \(q\)-element exists that does not occur in the list. This is done by collecting a second list badcases_thm of the remaining "bad" cases.
For the proof of the stronger version, we will later explicitly compute group elements from the classes in question that generate \(S\) together with a Sylow \(p\)-subgroup. (The only technical complication is that the class fusion of maximal subgroups of the type \((2^2 \times F_4(2)):2\) of the Baby Monster is currently not known, thus we cannot simply induce the trivial character in this case. However, the permutation character is uniquely determined by the two character tables.)
gap> badcases_thm:= [];; gap> badcases_strong:= [];; gap> for name in Filtered( names, x -> x <> "M" ) do > t:= CharacterTable( name ); > orders:= OrdersClassRepresentatives( t ); > n:= NrConjugacyClasses( t ); > mx:= List( Maxes( t ), CharacterTable ); > for p in PrimeDivisors( Size( t ) ) do > good:= Filtered( mx, s -> ( Size( t ) / Size( s ) ) mod p <> 0 ); > primperm:= []; > for s in good do > if GetFusionMap( s, t ) <> fail then > Add( primperm, TrivialCharacter( s )^t ); > else > ind:= Set( PossibleClassFusions( s, t ), > map -> InducedClassFunctionsByFusionMap( s, t, > [ TrivialCharacter( s ) ], map )[1] ); > if Length( ind ) <> 1 then > Error( "permutation character not uniquely determined" ); > fi; > Add( primperm, ind[1] ); > fi; > od; > sum:= Sum( [ 1 .. Length( good ) ], > i -> upper_bound( t, good[i], p ) > * primperm[i] / primperm[i][1] ); > badpos:= Filtered( [ 2 .. Length( sum ) ], i -> sum[i] >= 1 ); > if badpos <> [] then > Add( badcases_strong, [ name, p, ShallowCopy( badpos ) ] ); > for i in ShallowCopy( badpos ) do > q:= SmallestRootInt( orders[i] ); > if IsPrimeInt( q ) then > if ForAny( [ 2 .. n ], > j -> SmallestRootInt( orders[j] ) = q > and not j in badpos ) then > RemoveSet( badpos, i ); > fi; > fi; > od; > if not IsEmpty( badpos ) then > Add( badcases_thm, [ name, p, badpos ] ); > fi; > fi; > od; > od; gap> badcases_thm; [ [ "M23", 3, [ 3 ] ], [ "HS", 3, [ 4, 11 ] ] ] gap> badcases_strong; [ [ "M11", 5, [ 2 ] ], [ "M12", 5, [ 3, 4 ] ], [ "M22", 3, [ 2 ] ], [ "M22", 5, [ 2 ] ], [ "J2", 3, [ 2 ] ], [ "M23", 3, [ 2, 3 ] ], [ "M23", 5, [ 2 ] ], [ "M23", 7, [ 2 ] ], [ "HS", 3, [ 2, 3, 4, 5, 6, 7, 9, 11 ] ], [ "HS", 5, [ 2, 3, 5 ] ], [ "M24", 5, [ 2, 4 ] ], [ "M24", 7, [ 2, 4 ] ], [ "He", 5, [ 2 ] ], [ "Co2", 3, [ 2, 3 ] ], [ "Fi22", 5, [ 2 ] ], [ "Fi22", 7, [ 2 ] ], [ "Fi23", 5, [ 2, 3, 5 ] ], [ "Fi23", 7, [ 2 ] ], [ "B", 7, [ 2 ] ] ]
Most of these open cases can be ruled out by constructing the group \(S\) and a Sylow \(p\)-subgroup \(P\) in question and then finding explicit elements \(x\) such that \(S\) is generated by \(P\) and \(x\). For that, we use the data from the Atlas of Group Representations [WWT+].
The following function tries to find random elements from all conjugacy classes of nonidentity elements that have the desired property. It returns fail if no straight line program is available for computing class representatives, and returns \(P\) and the list of class representatives that generate together with \(P\) if such elements were found. Thus the function will not return if the generation property does not hold.
gap> prove_generation:= function( name, p ) > local S, prg, P, reps, good, x, g, U; > > prg:= AtlasProgram( name, "classes" ); > if prg = fail then > return fail; > fi; > > S:= AtlasGroup( name ); > P:= SylowSubgroup( S, p ); > reps:= ResultOfStraightLineProgram( prg.program, GeneratorsOfGroup( S ) ); > good:= []; > for x in Filtered( reps, x -> Order( x ) <> 1 ) do > repeat > g:= Random( S ); > U:= ClosureGroup( P, x^g ); > until Size( U ) = Size( S ); > Add( good, x^g ); > od; > > return [ P, good ]; > end;; gap> for entry in badcases_strong do > res:= prove_generation( entry[1], entry[2] ); > if res = fail then > Print( "no classes script for ", entry, "\n" ); > fi; > od; no classes script for [ "He", 5, [ 2 ] ] no classes script for [ "Fi22", 5, [ 2 ] ] no classes script for [ "Fi22", 7, [ 2 ] ] no classes script for [ "Fi23", 5, [ 2, 3, 5 ] ] no classes script for [ "Fi23", 7, [ 2 ] ] no classes script for [ "B", 7, [ 2 ] ]
In the remaining six cases, we show only the generation property for the class representatives in the list. These are involutions from the class 2A, and for the group \(Fi_{23}\) and \(p = 5\) additionally elements from the classes 2B and 3A.
A 2A element in the group \(He\) can be found as the fifth power of any element of order \(10\).
gap> S:= AtlasGroup( "He" );; gap> repeat > x:= Random( S ); > until Order( x ) = 10; gap> x:= x^5;; gap> P5:= SylowSubgroup( S, 5 );; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P5, x^g ); > until Size( U ) = Size( S );
A 2A element in the group \(Fi_{22}\) can be found as the \(15\)-th power of any element of order \(30\).
gap> S:= AtlasGroup( "Fi22" );; gap> repeat > x:= Random( S ); > until Order( x ) = 30; gap> x:= x^15;; gap> P5:= SylowSubgroup( S, 5 );; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P5, x^g ); > until Size( U ) = Size( S ); gap> P7:= SylowSubgroup( S, 7 );; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P7, x^g ); > until Size( U ) = Size( S );
A 2A element in the group \(Fi_{23}\) can be found as the \(21\)-st power of any element of order \(42\).
gap> S:= AtlasGroup( "Fi23" );; gap> repeat > x:= Random( S ); > until Order( x ) = 42; gap> x:= x^21;; gap> P5:= SylowSubgroup( S, 5 );; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P5, x^g ); > until Size( U ) = Size( S ); gap> P7:= SylowSubgroup( S, 7 );; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P7, x^g ); > until Size( U ) = Size( S );
A 2B element in the group \(Fi_{23}\) can be found as the \(30\)-th power of any element of order \(60\).
gap> repeat > x:= Random( S ); > until Order( x ) = 60; gap> x:= x^30;; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P5, x^g ); > until Size( U ) = Size( S );
A 3A element in the group \(Fi_{23}\) can be found as the \(20\)-th power of any element of order \(60\).
gap> repeat > x:= Random( S ); > until Order( x ) = 60; gap> x:= x^20;; gap> repeat > g:= Random( S ); > U:= ClosureGroup( P5, x^g ); > until Size( U ) = Size( S );
In the open case for the Baby Monster, we have to show that the group is generated by a 2A element and an element of order \(7\). This can be done character-theoretically, for example as follows. There are such elements \(x\) and \(y\) whose product \(x y\) has order \(47\), and the only proper subgroups of the Baby Monster that contain elements of order \(47\) are contained in maximal subgroups of the type \(47:23\). Thus \(x\) and \(y\) generate the Baby Monster.
gap> t:= CharacterTable( "B" );; gap> 7pos:= Positions( OrdersClassRepresentatives( t ), 7 ); [ 31 ] gap> 47pos:= Positions( OrdersClassRepresentatives( t ), 47 ); [ 172, 173 ] gap> ClassMultiplicationCoefficient( t, 2, 7pos[1], 47pos[1] ); 7332 gap> Filtered( Maxes( t ), > x -> Size( CharacterTable( x ) ) mod 47 = 0 ); [ "47:23" ]
Now consider the case that \(S\) is the Monster, which is special because the complete list of classes of maximal subgroups of \(S\) is currently not known. From [NW13] and [Wil] we know \(44\) classes of maximal subgroups, and that each possible additional maximal subgroup is almost simple and has socle \(L_2(13)\), \(U_3(4)\), \(U_3(8)\), or \(Sz(8)\). This implies that we know all those maximal subgroups that contain a Sylow-\(p\)-subgroup of \(S\) except in the case \(p = 19\), where maximal subgroups with socle \(U_3(8)\) may arise.
Thus let us first consider that at least one of \(p\), \(r\) is different from \(19\). In this situation, we use the same approach as for the other sporadic simple groups. The only complication is that not all permutation characters \(1_M^S\), for the relevant maximal subgroups \(M\) of \(S\), are known; however, if this happens then the character table of \(M\) is known, and we can compute the possible permutation characters, and take the common upper bounds for these characters. In each case, we get that the claimed property holds.
gap> t:= CharacterTable( "M" );; gap> orders:= OrdersClassRepresentatives( t );; gap> for p in Difference( PrimeDivisors( Size( t ) ), [ 19 ] ) do > goodpos:= Filtered( [ 1 .. Length( Monster_prim_data ) ], > i -> monstermaxindices[i] mod p <> 0 ); > sum:= ListWithIdenticalEntries( NrConjugacyClasses( t ), 0 ); > for i in goodpos do > if Length( Monster_prim_data[i] ) = 2 then > # We know the permutation character but not the subgroup table. > sum:= sum + upper_bound( t, fail, p ) > * Monster_prim_data[i][2] / monstermaxindices[i]; > else > s:= monstermaxtables[i]; > if GetFusionMap( s, t ) <> fail then > # We can compute the permutation character. > sum:= sum + upper_bound( t, s, p ) > * TrivialCharacter( s )^t / monstermaxindices[i]; > else > # We get only candidates for the permutation character. > cand:= Set( PossibleClassFusions( s, t ), > map -> InducedClassFunctionsByFusionMap( s, t, > [ TrivialCharacter( s ) ], map )[1] ); > # For each class, take the maximum of the possible values. > sum:= sum + upper_bound( t, s, p ) > * List( TransposedMat( cand ), Maximum ) > / monstermaxindices[i]; > fi; > fi; > od; > badpos:= Filtered( [ 2 .. Length( sum ) ], i -> sum[i] >= 1 ); > if badpos <> [] then > Error( "check open cases in ", badpos, "\n" ); > fi; > od;
Finally, let \(p = r = 19\). The group \(S\) has exactly one class of elements of order \(19\). Let \(x\) be such an element. From the character table of \(S\), we compute that there exist conjugates \(y\) of \(x\) such that \(x y\) has order \(71\). Since \(\langle x, y \rangle = \langle x, x y \rangle\) holds and no maximal subgroup of \(S\) has order divisible by \(19 \cdot 71\), we have \(\langle x, y \rangle = S\).
gap> pos19:= Positions( OrdersClassRepresentatives( t ), 19 ); [ 63 ] gap> pos71:= Positions( OrdersClassRepresentatives( t ), 71 ); [ 169, 170 ] gap> ClassMultiplicationCoefficient( t, pos19[1], pos19[1], pos71[1] ); 621743152370566020417806353602387433415016198936 gap> ForAny( monstermaxindices, > x -> ( Size( t ) / x ) mod ( 19 * 71 ) = 0 ); false gap> ForAny( [ "L2(13)", "U3(4)", "U3(8)", "Sz(8)" ], > x -> Size( CharacterTable( x ) ) mod 71 = 0 ); false
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