This chapter shows a few examples how to use the functionality provided by the CTBlocks package.
The aim of this section is to produce an overview of the non-cyclic, abelian, faithful blocks of sporadic simple groups and their covers, in a similar format as the tables that are available at http://www.math.rwth-aachen.de/~Felix.Noeske/tabular.pdf.
The function DisplayBlockInvariants (3.10-4) can be used to show an overview of block invariants.
(One short remark is necessary in advance. Depending on the terminal capabilities, the borders of the table printed by DisplayBlockInvariants (3.10-4) may contain non-ASCII characters. However, these characters are not supported by the LaTeX and HTML versions of GAPDoc documents. For the examples in this section, we set the user preference DisplayFunction (see Section AtlasRep: User preference DisplayFunction) to the value "Print", in order to produce output consisting only of ASCII characters.)
gap> origpref:= UserPreference( "AtlasRep", "DisplayFunction" );; gap> SetUserPreference( "AtlasRep", "DisplayFunction", "Print" );
The simplest way to use DisplayBlockInvariants (3.10-4) is to call it with the identifier of an ordinary character table from GAP's library of character tables, for example the table of the double cover of the alternating group \(A_6\) on six points.
gap> DisplayBlockInvariants( "2.A6" ); Block invariants for 2.A6 ---------------------------------------------- | p | b | d | k | l | c | a | n | f | r | sr | ---------------------------------------------- | 2 | 1 | 4 | 9 | 3 | - | - | - | + | + | + | | | 2 | 1 | 2 | 1 | + | + | + | + | + | - | | | 3 | 1 | 2 | 1 | + | + | + | + | + | - | | 3 | 1 | 2 | 6 | 4 | - | + | - | - | + | + | | | 2 | 0 | 1 | 1 | + | + | + | - | + | - | | | 3 | 2 | 6 | 4 | - | + | - | + | + | + | | 5 | 1 | 1 | 4 | 2 | + | + | - | - | + | + | | | 2 | 0 | 1 | 1 | + | + | + | - | + | - | | | 3 | 0 | 1 | 1 | + | + | + | - | + | - | | | 4 | 0 | 1 | 1 | + | + | + | - | + | - | | | 5 | 1 | 4 | 2 | + | + | - | + | + | + | | | 6 | 0 | 1 | 1 | + | + | + | + | + | - | | | 7 | 0 | 1 | 1 | + | + | + | + | + | - | ----------------------------------------------
The rows of the table correspond to the \(p\)-blocks of the table, for all prime divisors \(p\) of the group order. The columns of the table are labelled by
p, b, d: the characteristic, the block number, and the defect of the block,
k and l: the numbers of ordinary and modular irreducibles in the block,
c, a, n: information whether the defect groups of the block are cyclic, abelian, and normal, respectively,
f, r, sr: information whether the block is faithful, real, and strongly real, respectively.
We are interested in an overview that is on the one hand more general in the sense that several groups at handled at the same time, and which is on the other hand restricted to faithful blocks with abelian but noncyclic defect groups. As an example, we take the central extensions of the simple group \(Fi_{22}\). (Here we fetch the character tables in the beginning. As we will see below, we will call the relevant functions a second time, and if we use the same character table object in the second round, the values in question are already known and can just be fetched.)
gap> tbls:= List( [ "Fi22", "2.Fi22", "3.Fi22", "6.Fi22" ], > CharacterTable );; gap> DisplayBlockInvariants( tbls, IsFaithful, true, > IsBlockWithCyclicDefectGroup, false, > IsBlockWithAbelianDefectGroup, [ true, fail ] ); Block invariants faithful noncyclic defect group --------------------------------------------------- | G | p | b | d | k | l | a | n | r | sr | --------------------------------------------------- | Fi22 | 5 | 1 | 2 | 20 | 16 | + | - | + | + | | 2.Fi22 | 2 | 3 | 2 | 4 | 1 | + | - | + | + | | | 5 | 39 | 2 | 20 | 16 | + | - | + | + | | 3.Fi22 | 3 | 2 | 2 | 9 | 2 | + | - | + | + | | | | 3 | 2 | 9 | 2 | + | - | + | + | | | 5 | 39 | 2 | 20 | 16 | + | - | - | - | | | | 40 | 2 | 20 | 16 | + | - | - | - | | 6.Fi22 | 5 | 107 | 2 | 20 | 16 | + | - | - | - | | | | 108 | 2 | 20 | 16 | + | - | - | - | ---------------------------------------------------
Note that we can in general not decide from the character table whether the defect groups are abelian, therefore we have to allow for the possible values true and fail as results of IsBlockWithAbelianDefectGroup (3.4-10).
This result is not yet good enough: We want to show a separate table for each prime \(p\) that divides the order of one of the groups. (And we do not want to print the header information for each prime.)
gap> primes:= Union( List( tbls, t -> Set( Factors( Size( t ) ) ) ) ); [ 2, 3, 5, 7, 11, 13 ] gap> for p in primes do > DisplayBlockInvariants( tbls, IsFaithful, true, > IsBlockWithCyclicDefectGroup, false, > IsBlockWithAbelianDefectGroup, [ true, fail ], > UnderlyingCharacteristic, p, > rec( header:= [ "", Concatenation( "p = ", String( p ) ) ] ) ); > od; p = 2 ------------------------------------------- | G | b | d | k | l | a | n | r | sr | ------------------------------------------- | 2.Fi22 | 3 | 2 | 4 | 1 | + | - | + | + | ------------------------------------------- p = 3 ------------------------------------------- | G | b | d | k | l | a | n | r | sr | ------------------------------------------- | 3.Fi22 | 2 | 2 | 9 | 2 | + | - | + | + | | | 3 | 2 | 9 | 2 | + | - | + | + | ------------------------------------------- p = 5 ----------------------------------------------- | G | b | d | k | l | a | n | r | sr | ----------------------------------------------- | Fi22 | 1 | 2 | 20 | 16 | + | - | + | + | | 2.Fi22 | 39 | 2 | 20 | 16 | + | - | + | + | | 3.Fi22 | 39 | 2 | 20 | 16 | + | - | - | - | | | 40 | 2 | 20 | 16 | + | - | - | - | | 6.Fi22 | 107 | 2 | 20 | 16 | + | - | - | - | | | 108 | 2 | 20 | 16 | + | - | - | - | -----------------------------------------------
Now we turn to all sporadic simple groups and their central extensions. First we collect the names of the character tables, in the same order as we need for the overview, that is, according to increasing order of the simple groups.
gap> names:= [ "M11", "M12", "2.M12", "J1", "M22", "2.M22", > "3.M22", "4.M22", "6.M22", "12.M22", "J2", "2.J2", > "M23", "HS", "2.HS", "J3", "3.J3", "M24", "McL", "3.McL", > "He", "Ru", "2.Ru", "Suz", "2.Suz", "3.Suz", "6.Suz", > "ON", "3.ON", "Co3", "Co2", "Fi22", "2.Fi22", "3.Fi22", > "6.Fi22", "HN", "Ly", "Th", "Fi23", "Co1", "2.Co1", > "J4", "F3+", "3.F3+", "B", "2.B", "M" ];; gap> tbls:= List( names, CharacterTable );;
The overview is printed in the same way as above.
gap> primes:= Union( List( tbls, t -> Set( Factors( Size( t ) ) ) ) );; gap> for p in primes do > DisplayBlockInvariants( tbls, IsFaithful, true, > IsBlockWithCyclicDefectGroup, false, > IsBlockWithAbelianDefectGroup, [ true, fail ], > UnderlyingCharacteristic, p, > rec( header:= [ "", Concatenation( "p = ", String( p ) ) ] ) ); > od; p = 2 ------------------------------------------- | G | b | d | k | l | a | n | r | sr | ------------------------------------------- | M12 | 2 | 2 | 4 | 3 | + | - | + | + | | J1 | 1 | 3 | 8 | 5 | + | - | + | + | | J2 | 2 | 2 | 4 | 3 | + | - | + | + | | HS | 2 | 2 | 4 | 3 | + | - | + | + | | Ru | 2 | 2 | 4 | 3 | + | - | + | + | | Co3 | 2 | 3 | 8 | 5 | + | - | + | + | | 2.Fi22 | 3 | 2 | 4 | 1 | + | - | + | + | | F3+ | 2 | 2 | 4 | 3 | + | - | + | + | ------------------------------------------- p = 3 ---------------------------------------------- | G | b | d | k | l | a | n | r | sr | ---------------------------------------------- | M11 | 1 | 2 | 9 | 7 | + | - | + | + | | M22 | 1 | 2 | 6 | 5 | + | - | + | + | | 2.M22 | 6 | 2 | 6 | 5 | + | - | + | + | | 3.M22 | 2 | 2 | 9 | 2 | + | - | + | + | | 4.M22 | 10 | 2 | 6 | 5 | + | - | - | - | | | 11 | 2 | 6 | 5 | + | - | - | - | | 6.M22 | 7 | 2 | 9 | 2 | + | - | + | + | | M23 | 1 | 2 | 9 | 7 | + | - | + | + | | HS | 1 | 2 | 9 | 7 | + | - | + | + | | | 2 | 2 | 9 | 7 | + | - | + | + | | 2.HS | 7 | 2 | 9 | 5 | + | - | + | + | | 3.J3 | 2 | 2 | 9 | 2 | + | - | + | + | | He | 2 | 2 | 9 | 7 | + | - | + | + | | Suz | 2 | 2 | 6 | 5 | + | - | + | + | | 3.Suz | 3 | 2 | 9 | 2 | + | - | + | + | | ON | 1 | 4 | 18 | 14 | + | - | + | + | | | 2 | 2 | 6 | 5 | + | - | + | + | | 3.Fi22 | 2 | 2 | 9 | 2 | + | - | + | + | | | 3 | 2 | 9 | 2 | + | - | + | + | | HN | 2 | 2 | 9 | 7 | + | - | + | + | | Co1 | 3 | 2 | 9 | 5 | + | - | + | + | | J4 | 6 | 2 | 9 | 5 | + | - | + | + | | F3+ | 2 | 2 | 6 | 4 | + | - | + | + | | B | 2 | 2 | 9 | 7 | + | - | + | + | | | 3 | 2 | 9 | 7 | + | - | + | + | | | 6 | 2 | 9 | 5 | + | - | + | + | ---------------------------------------------- p = 5 ----------------------------------------------- | G | b | d | k | l | a | n | r | sr | ----------------------------------------------- | J2 | 1 | 2 | 14 | 6 | + | - | + | + | | 2.J2 | 6 | 2 | 14 | 6 | + | - | + | + | | He | 1 | 2 | 16 | 14 | + | - | + | + | | Suz | 1 | 2 | 16 | 12 | + | - | + | + | | 2.Suz | 19 | 2 | 16 | 12 | + | - | + | + | | 3.Suz | 18 | 2 | 16 | 12 | + | - | - | - | | | 19 | 2 | 16 | 12 | + | - | - | - | | 6.Suz | 59 | 2 | 16 | 12 | + | - | - | - | | | 60 | 2 | 16 | 12 | + | - | - | - | | Fi22 | 1 | 2 | 20 | 16 | + | - | + | + | | 2.Fi22 | 39 | 2 | 20 | 16 | + | - | + | + | | 3.Fi22 | 39 | 2 | 20 | 16 | + | - | - | - | | | 40 | 2 | 20 | 16 | + | - | - | - | | 6.Fi22 | 107 | 2 | 20 | 16 | + | - | - | - | | | 108 | 2 | 20 | 16 | + | - | - | - | | Fi23 | 1 | 2 | 20 | 16 | + | - | + | + | | | 2 | 2 | 20 | 16 | + | - | + | + | | Co1 | 3 | 2 | 16 | 12 | + | - | + | + | | F3+ | 1 | 2 | 20 | 16 | + | - | + | + | | | 2 | 2 | 16 | 14 | + | - | + | + | | | 3 | 2 | 20 | 16 | + | - | + | + | | 3.F3+ | 45 | 2 | 20 | 16 | + | - | - | - | | | 46 | 2 | 20 | 16 | + | - | - | - | | | 47 | 2 | 20 | 14 | + | - | - | - | | | 48 | 2 | 20 | 14 | + | - | - | - | | B | 2 | 2 | 20 | 16 | + | - | + | + | | | 8 | 2 | 20 | 16 | + | - | + | + | | M | 4 | 2 | 20 | 16 | + | - | + | + | ----------------------------------------------- p = 7 --------------------------------------------- | G | b | d | k | l | a | n | r | sr | --------------------------------------------- | Th | 1 | 2 | 27 | 24 | + | - | + | + | | Co1 | 1 | 2 | 27 | 21 | + | - | + | + | | 2.Co1 | 46 | 2 | 27 | 21 | + | - | + | + | | B | 1 | 2 | 27 | 24 | + | - | + | + | | | 2 | 2 | 27 | 24 | + | - | + | + | | | 4 | 2 | 27 | 21 | + | - | + | + | | 2.B | 73 | 2 | 27 | 24 | + | - | + | + | | M | 2 | 2 | 27 | 24 | + | - | + | + | --------------------------------------------- p = 11 ---------------------------------------- | G | b | d | k | l | a | n | r | sr | ---------------------------------------- | M | 1 | 2 | 50 | 45 | + | - | + | + | ----------------------------------------
Note:
The two \(2\)-blocks of defect two for \(12.M_{22}\) which are listed in http://www.math.rwth-aachen.de/~Felix.Noeske/tabular.pdf have a cyclic defect group (the central subgroup of order four) and are therefore not contained in the above output.
The principal \(2\)-block of \(J_1\) which appears in the above output has been deliberately omitted from the table available in the web.
Finally, we reset the user preference.
gap> SetUserPreference( "AtlasRep", "DisplayFunction", origpref );
We have seen in Section 2.1 that the character tables of the sporadic simple groups and their central extensions determine for all their \(p\)-blocks whether the defect groups are abelian: Only - entries occur in the column a of the tables shown in that section.
In general this is not the case. Let us look for an example where IsBlockWithAbelianDefectGroup (3.4-10) returns fail. (This computation takes several minutes.)
gap> b:= OnePBlock( AllCharacterTableNames(), > IsBlockWithAbelianDefectGroup, fail ); Block( CharacterTable( "Isoclinic(2x2.F4(2).2)" ), 2, 2 ) gap> Defect( b ); 3 gap> dcl:= ClassPositionsOfDefectGroupOfBlock( b ); [ 1, 2, 3, 4, 215, 216 ] gap> t:= UnderlyingCharacterTable( b );; gap> OrdersClassRepresentatives( t ){ dcl }; [ 1, 4, 2, 4, 2, 4 ]
In this example, each defect group in question has order eight and contains at least two involutions and at least three elements of order four. This excludes the two types of nonabelian groups of order eight, hence the defect groups are abelian.
We see that we could easily extend IsBlockWithAbelianDefectGroup (3.4-10) such that true is returned in the above situation. Similarly, the lattice of normal subgroups can be used in some cases to conclude that the defect groups in question are abelian. However, there are also examples of simple groups for which IsBlockWithAbelianDefectGroup (3.4-10) fails.
gap> b:= OnePBlock( AllCharacterTableNames( IsSimple, true ), > IsBlockWithAbelianDefectGroup, fail ); Block( CharacterTable( "L8(2)" ), 3, 5 ) gap> Defect( b ); 3
Block induction in the sense of Brauer (see BrauerCorrespondent (3.8-1)) implies \(p\)-regular block induction (see PRegularCorrespondent (3.8-2)), which implies extended block induction (see WheelerCorrespondent (3.8-4)), and also block induction in the sense of Alperin-Burry (see AlperinBurryCorrespondent (3.8-3)) implies extended block induction.
We list those examples from [BH01] that separate these concepts of block induction.
A series of examples where block induction in the sense of Brauer is defined but block induction in the sense of Alperin-Burry is not defined is given in [BH01, Example 2.2]. The smallest member of this series is the dihedral group \(G\) of order \(24\) and \(H\) is its \(2\)-core (a cyclic group of order four).
gap> g:= DihedralGroup( 24 );; gap> p:= 2;; gap> t:= CharacterTable( g );; gap> h:= PCore( g, 2 );; gap> Size( h ); 4 gap> s:= CharacterTable( h );; gap> AlperinBurryCorrespondent( s, t, p, 1 ); fail gap> BrauerCorrespondent( s, t, p, 1 ); 1 gap> PRegularCorrespondent( s, t, p, 1 ); 1 gap> WheelerCorrespondent( s, t, p, 1 ); 1
An example where block induction in the sense of Brauer and block induction in the sense of Alperin-Burry are not defined but \(p\)-regular block induction is defined is given by \(G = S_3\), the symmetric group on three points, and \(H\) its trivial subgroup, for \(p = 2\).
gap> g:= SymmetricGroup( 3 );; gap> p:= 2;; gap> t:= CharacterTable( g );; gap> h:= TrivialSubgroup( g );; gap> s:= CharacterTable( h );; gap> AlperinBurryCorrespondent( s, t, p, 1 ); fail gap> BrauerCorrespondent( s, t, p, 1 ); fail gap> PRegularCorrespondent( s, t, p, 1 ); 1 gap> WheelerCorrespondent( s, t, p, 1 ); 1
A series of examples where \(p\)-regular block induction is not defined but block induction in the sense of Alperin-Burry is defined is given in [BH01, Example 2.3]. One instance of this series is the alternating group \(G\) on four points where \(H\) has order three and \(p = 2\) holds.
gap> g:= AlternatingGroup( 4 );; gap> p:= 2;; gap> t:= CharacterTable( g );; gap> h:= SylowSubgroup( g, 3 );; gap> s:= CharacterTable( h );; gap> AlperinBurryCorrespondent( s, t, p, 1 ); 1 gap> BrauerCorrespondent( s, t, p, 1 ); fail gap> PRegularCorrespondent( s, t, p, 1 ); fail gap> WheelerCorrespondent( s, t, p, 1 ); 1
An example where \(p\)-regular block induction and block induction in the sense of Alperin-Burry are defined but block induction in the sense of Brauer is not defined is given by the cyclic group \(G\) of order \(p = 2\) and \(H\) its trivial subgroup.
gap> g:= CyclicGroup( 2 );; gap> p:= 2;; gap> t:= CharacterTable( g );; gap> h:= TrivialSubgroup( g );; gap> s:= CharacterTable( h );; gap> AlperinBurryCorrespondent( s, t, p, 1 ); 1 gap> BrauerCorrespondent( s, t, p, 1 ); fail gap> PRegularCorrespondent( s, t, p, 1 ); 1 gap> WheelerCorrespondent( s, t, p, 1 ); 1
An example where extended block induction is defined but neither \(p\)-regular block induction nor block induction in the sense of Alperin-Burry are defined is given by [Whe94, Example 2.10]. We have \(G\) the simple group of order \(168\), \(p = 3\), and \(H\) is a Sylow \(2\)-subgroup of \(G\).
gap> g:= PSL(2,7);; gap> Size( g ); IsSimple( g ); 168 true gap> p:= 3;; gap> t:= CharacterTable( g );; gap> h:= SylowSubgroup( g, 2 );; gap> s:= CharacterTable( h );; gap> AlperinBurryCorrespondent( s, t, p, 1 ); fail gap> BrauerCorrespondent( s, t, p, 1 ); fail gap> PRegularCorrespondent( s, t, p, 1 ); fail gap> WheelerCorrespondent( s, t, p, 1 ); 4
(The smallest such example is the symmetric group \(G\) on three points, its trivial subgroup \(H\), and \(p = 2\).)
We verify the results of [LP10, Example 4.7.8].
First we show that for \(p = 2\), block induction in the sense of Brauer (see BrauerCorrespondent (3.8-1)) is defined from a \(p\)-block \(b\) of a proper subgroup \(H\) of the Mathieu group \(M_{11}\) if and only if \(H\) has even order and \(b\) is the principal block of \(H\).
gap> tom:= TableOfMarks( "M11" );; gap> g:= UnderlyingGroup( tom );; gap> p:= 2;; gap> good:= [];; gap> bad:= [];; gap> for i in [ 1 .. Length( OrdersTom( tom ) ) - 1 ] do > h:= RepresentativeTom( tom, i ); > tblh:= CharacterTable( h ); > for b in [ 1 .. Length( PrimeBlocks( tblh, p ).defect ) ] do > if BrauerCorrespondent( tblh, CharacterTable( g ), p, b ) <> fail then > AddSet( good, [ Size( h ), b ] ); > else > AddSet( bad, [ Size( h ), b ] ); > fi; > od; > od; gap> good; [ [ 2, 1 ], [ 4, 1 ], [ 6, 1 ], [ 8, 1 ], [ 10, 1 ], [ 12, 1 ], [ 16, 1 ], [ 18, 1 ], [ 20, 1 ], [ 24, 1 ], [ 36, 1 ], [ 48, 1 ], [ 60, 1 ], [ 72, 1 ], [ 120, 1 ], [ 144, 1 ], [ 360, 1 ], [ 660, 1 ], [ 720, 1 ] ] gap> bad; [ [ 1, 1 ], [ 3, 1 ], [ 3, 2 ], [ 3, 3 ], [ 5, 1 ], [ 5, 2 ], [ 5, 3 ], [ 5, 4 ], [ 5, 5 ], [ 6, 2 ], [ 6, 3 ], [ 9, 1 ], [ 9, 2 ], [ 9, 3 ], [ 9, 4 ], [ 9, 5 ], [ 9, 6 ], [ 9, 7 ], [ 9, 8 ], [ 9, 9 ], [ 10, 2 ], [ 10, 3 ], [ 11, 1 ], [ 11, 2 ], [ 11, 3 ], [ 11, 4 ], [ 11, 5 ], [ 11, 6 ], [ 11, 7 ], [ 11, 8 ], [ 11, 9 ], [ 11, 10 ], [ 11, 11 ], [ 12, 2 ], [ 18, 2 ], [ 18, 3 ], [ 18, 4 ], [ 18, 5 ], [ 18, 6 ], [ 20, 2 ], [ 36, 2 ], [ 36, 3 ], [ 36, 4 ], [ 55, 1 ], [ 55, 2 ], [ 55, 3 ], [ 55, 4 ], [ 55, 5 ], [ 55, 6 ], [ 55, 7 ], [ 60, 2 ], [ 72, 2 ], [ 72, 3 ], [ 120, 2 ], [ 144, 2 ], [ 360, 2 ], [ 360, 3 ], [ 660, 2 ], [ 660, 3 ], [ 660, 4 ], [ 720, 2 ] ]
Next we inspect for which \(2\)-blocks of cyclic subgroups of order six in the Mathieu group \(M_{12}\), block induction in the sense of Brauer is defined.
gap> tom:= TableOfMarks( "M12" );; gap> g:= UnderlyingGroup( tom );; gap> p:= 2;; gap> results:= [];; gap> for i in Positions( OrdersTom( tom ), 6 ) do > h:= RepresentativeTom( tom, i ); > if IsCyclic( h ) then > tblh:= CharacterTable( h ); > Add( results, List( [ 1 .. Length( PrimeBlocks( tblh, p ).defect ) ], > b -> BrauerCorrespondent( tblh, CharacterTable( g ), p, b ) ) ); > fi; > od; gap> results; [ [ 1, fail, fail ], [ fail, 2, 2 ] ]
We see that there are two classes of such subgroups in \(M_{12}\), and block induction is defined exactly for the principal block of the groups in the first class, whereas block induction is defined exactly for the two non-principal blocks of the groups in the second class.
We compute which real classes of sporadic simple groups are not strongly real.
gap> names:= AllCharacterTableNames( IsSporadicSimple, true, > IsDuplicateTable, false ); [ "B", "Co1", "Co2", "Co3", "F3+", "Fi22", "Fi23", "HN", "HS", "He", "J1", "J2", "J3", "J4", "Ly", "M", "M11", "M12", "M22", "M23", "M24", "McL", "ON", "Ru", "Suz", "Th" ] gap> for name in names do > t:= CharacterTable( name ); > test:= Filtered( [ 1 .. NrConjugacyClasses( t ) ], > i -> IsRealClass( t, i ) > and not IsStronglyRealClass( t, i ) ); > if test <> [] then > Print( name, ": ", ClassNames( t, "Atlas" ){ test }, "\n" ); > fi; > od; Co2: [ "16B" ] HN: [ "8A" ] M: [ "8C", "8F", "24F", "24G", "24H", "24J", "32A", "32B", "40A", "48A" ] M22: [ "8A" ] M23: [ "8A" ] McL: [ "3A", "5A", "6A", "8A", "10A", "12A" ] Th: [ "8B" ]
The article [Sul08] claims to list the real classes in sporadic simple groups that are not strongly real. However, the classes of element order \(8\), \(10\), and \(12\) in \(McL\) are missing in that list, and the classes 16AB, 22BC, 23AB of \(Fi_{23}\) are erroneously claimed to be real.
Let us look at these cases.
The group \(McL\) has a unique class of involutions, and only elements of order up to \(6\) are products of at most two involutions. On the other hand, six more classes of \(McL\) are real.
gap> t:= CharacterTable( "McL" ); CharacterTable( "McL" ) gap> orders:= OrdersClassRepresentatives( t ); [ 1, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 8, 9, 9, 10, 11, 11, 12, 14, 14, 15, 15, 30, 30 ] gap> nccl:= Length( orders ); 24 gap> Filtered( [ 1 .. nccl ], > i -> ClassMultiplicationCoefficient( t, 2, 2, i ) <> 0 ); [ 1, 2, 4, 5, 7, 9 ] gap> Filtered( [ 1 .. nccl ], i -> PowerMap( t, -1, i ) = i ); [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 18 ]
The classes 16AB, 22BC, 23AB of \(Fi_{23}\) are not real.
gap> t:= CharacterTable( "Fi23" ); CharacterTable( "Fi23" ) gap> orders:= OrdersClassRepresentatives( t );; gap> filt:= PositionsProperty( orders, x -> x in [ 16, 22, 23 ] ); [ 63, 64, 77, 78, 79, 80, 81 ] gap> PowerMap( t, -1 ){ filt }; [ 64, 63, 77, 79, 78, 81, 80 ] gap> ClassNames( t, "Atlas" ){ filt }; [ "16A", "16B", "22A", "22B", "22C", "23A", "23B" ]
(Note that the irrational values in the six classes lie in quadratic number fields. For the element orders \(22\) and \(23\), it is clear from this fact that the classes cannot be real, because the fields of \(11\)-th and \(23\)-rd roots of unity do not have a real quadratic subfield.)
We compute which real \(p\)-blocks of sporadic simple groups are not strongly real.
gap> names:= AllCharacterTableNames( IsSporadicSimple, true, > IsDuplicateTable, false ); [ "B", "Co1", "Co2", "Co3", "F3+", "Fi22", "Fi23", "HN", "HS", "He", "J1", "J2", "J3", "J4", "Ly", "M", "M11", "M12", "M22", "M23", "M24", "McL", "ON", "Ru", "Suz", "Th" ] gap> bl:= AllPBlocks( names, IsRealBlock, true, > IsStronglyRealBlock, false ); [ Block( CharacterTable( "McL" ), 2, 4 ) ]
We get exactly one example, a \(2\)-block of defect one for the McLaughlin group \(McL\).
gap> Defect( bl[1] ); 1 gap> dcl:= ClassPositionsOfDefectClasses( bl[1] ); [ 6, 10, 11, 21, 22 ] gap> tbl:= UnderlyingCharacterTable( bl[1] );; gap> ClassNames( tbl, "Atlas" ){ dcl }; [ "5A", "7A", "7B", "15A", "15B" ] gap> Filtered( dcl, c -> IsRealClass( tbl, c ) ); [ 6 ] gap> Filtered( dcl, c -> IsStronglyRealClass( tbl, c ) ); [ ]
This block has exactly one real defect class, which is not strongly real. (The existence of a real defect class for each \(2\)-block is guaranteed by [GM00].)
For an algebra \(A\), let \(J(A)\) denote the Jacobson radical of \(A\), that is, the largest nilpotent ideal of \(A\). We consider the chain \(A = J(A)^0 \supset J(A) \supset J(A)^2 \supset \cdots \supset J(A)^{{n-1}} \supset J(A)^n = \{ 0 \}\) of ideals. Then \(n\) is called the Loewy length of \(A\). The factors \(J^{{i-1}} / J^i\), for \(1 \leq i \leq n\), are called the Loewy layers of \(A\).
The following functions compute, for a given algebra \(A\) and its radical \(J = J(A)\), the list of dimensions of the radical powers \(J(A)^i\) and of the Loewy layers of \(A\), respectively.
gap> DimensionsOfRadicalPowers:= function( A, J ) > local l, Jpower; > > l:= [ Dimension( J ) ]; > Jpower:= J; > while Dimension( Jpower ) <> 0 do > Jpower:= ProductSpace( Jpower, J ); > Add( l, Dimension( Jpower ) ); > od; > > return l; > end;; gap> DimensionsOfLoewyLayers:= function( A, J ) > local dims, l, i; > > if Dimension( A ) = 0 then > return []; > fi; > > dims:= DimensionsOfRadicalPowers( A, J ); > l:= [ Dimension( A ) - dims[1] ]; > for i in [ 2 .. Length( dims ) ] do > l[i]:= dims[ i-1 ] - dims[i]; > od; > > return l; > end;;
We are interested in the dimensions of the Loewy layers of the centres of the principal \(p\)-blocks of group algebras \(F G\), where \(G\) is a sporadic simple group and \(F\) is a field of characteristic \(p\). We are also interested in the dimensions of the Loewy layers of the centre of (the principal block of) \(F N_G(P)\), where \(P\) is a Sylow \(p\)-subgroup of \(G\).
(Note that these dimensions do not depend on the field \(F\), as long as the structure constants of the algebras live in \(F\). In our situation, we may choose \(F\) as a prime field, since the block idempotent of a principal block has always rational coefficients.)
Most of the relevant information is available in GAP's library of character tables, thus we will use the character theoretic variant of SCAlgebraCentreOfBlock (3.11-3).
Several normalizers of Sylow \(p\)-subgroups in sporadic simple groups have hundreds of conjugacy classes, and the computation of the radical of an algebra of that dimension is time consuming. Therefore, we use the fact that the Jacobson radical of the centre \(Z(F G)\) of an indecomposable group algebra \(F G\) is equal to the augmentation ideal of \(Z(F G)\) and thus has an \(F\)-basis consisting of the elements \(C^+ - |C| 1\), where \(C\) runs over the nonidentity conjugacy classes of \(G\), and \(C^+\) is the sum of the elements of \(C\). The following function can be used for that.
gap> RadicalOfIndecomposableGroupAlgebra:= function( A, tbl ) > local radgens, gens, classlengths, i; > > radgens:= []; > gens:= BasisVectors( CanonicalBasis( A ) ); > classlengths:= SizesConjugacyClasses( tbl ); > for i in [ 2 .. Length( gens ) ] do > Add( radgens, gens[i] - classlengths[i] * gens[1] ); > od; > > return SubalgebraNC( A, radgens, "basis" ); > end;;
What we want to compute is done by the following function. It assumes that the (ordinary) character table of the group \(G\) is available as tbl.
gap> DimensionsOfLoewyLayersByTable:= function( tbl, p ) > local A, J; > > if Length( PrimeBlocks( tbl, p ).defect ) = 1 then > # The group algebra is indecomposable. > A:= SCAlgebraCentreOfGroupAlgebra( tbl, p ); > J:= RadicalOfIndecomposableGroupAlgebra( A, tbl ); > else > # Use the simpleminded variant. > A:= SCAlgebraCentreOfBlock( tbl, p, 1 ); > J:= RadicalOfAlgebra( A ); > fi; > return DimensionsOfLoewyLayers( A, J ); > end;;
Some of these computations have been carried out in [Sch16], with different methods, and [Sch16, Conjecturere 5.8.1] states that for any sporadic simple group \(G\) and any prime \(p\) dividing \(|G|\), the Loewy length of \(Z(B)\) is expected to be an upper bound for the Loewy length of \(Z(F N_G(P))\), where \(B\) is the principal block of \(F G\), \(F\) is a field of characteristic \(p\), and \(P\) is a Sylow \(p\)-subgroup of \(G\). We check this conjecture as far as the available data admit this.
gap> conjecture_is_wrong_for:= [];; gap> conjecture_is_open_for:= [];; 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> for name in names do > Print( name, ":\n" ); > t:= CharacterTable( name ); > for p in Set( Factors( Size( t ) ) ) do > dimst:= DimensionsOfLoewyLayersByTable( t, p ); > Print( " ", String( p, 2 ), ": ", String( dimst, -30 ) ); > nname:= Concatenation( name, "N", String( p ) ); > n:= CharacterTable( nname ); > if n = fail then > Print( "(no Sylow normalizer table)\n" ); > Add( conjecture_is_open_for, [ name, p ] ); > else > dimsn:= DimensionsOfLoewyLayersByTable( n, p ); > if dimsn = dimst then > Print( "same dimensions for G and N_G(P)\n" ); > else > Print( dimsn, "\n" ); > fi; > # Check the conjecture. > if Length( dimst ) < Length( dimsn ) then > Add( conjecture_is_wrong_for, [ name, p ] ); > fi; > fi; > od; > od; M11: 2: [ 1, 3, 2, 1, 1 ] [ 1, 4, 1, 1 ] 3: [ 1, 7, 1 ] same dimensions for G and N_G(P) 5: [ 1, 4 ] same dimensions for G and N_G(P) 11: [ 1, 5, 1 ] same dimensions for G and N_G(P) M12: 2: [ 1, 7, 2, 1 ] [ 1, 14, 1 ] 3: [ 1, 8, 2 ] [ 1, 9, 1 ] 5: [ 1, 4 ] same dimensions for G and N_G(P) 11: [ 1, 5, 1 ] same dimensions for G and N_G(P) J1: 2: [ 1, 7 ] same dimensions for G and N_G(P) 3: [ 1, 2 ] same dimensions for G and N_G(P) 5: [ 1, 2, 1 ] same dimensions for G and N_G(P) 7: [ 1, 6 ] same dimensions for G and N_G(P) 11: [ 1, 10 ] same dimensions for G and N_G(P) 19: [ 1, 6, 1, 1 ] same dimensions for G and N_G(P) M22: 2: [ 1, 9, 2 ] [ 1, 16 ] 3: [ 1, 5 ] same dimensions for G and N_G(P) 5: [ 1, 4 ] same dimensions for G and N_G(P) 7: [ 1, 3, 1 ] same dimensions for G and N_G(P) 11: [ 1, 5, 1 ] same dimensions for G and N_G(P) J2: 2: [ 1, 12, 3, 1 ] [ 1, 12, 6 ] 3: [ 1, 8, 4 ] [ 1, 9, 3 ] 5: [ 1, 4, 5, 3, 1 ] same dimensions for G and N_G(P) 7: [ 1, 6 ] same dimensions for G and N_G(P) M23: 2: [ 1, 12, 2 ] [ 1, 16 ] 3: [ 1, 7, 1 ] same dimensions for G and N_G(P) 5: [ 1, 4 ] same dimensions for G and N_G(P) 7: [ 1, 3, 1 ] same dimensions for G and N_G(P) 11: [ 1, 5, 1 ] same dimensions for G and N_G(P) 23: [ 1, 11, 1 ] same dimensions for G and N_G(P) HS: 2: [ 1, 13, 4, 1, 1 ] [ 1, 30, 4 ] 3: [ 1, 7, 1 ] same dimensions for G and N_G(P) 5: [ 1, 13, 3 ] [ 1, 14, 2 ] 7: [ 1, 6 ] same dimensions for G and N_G(P) 11: [ 1, 5, 1 ] same dimensions for G and N_G(P) J3: 2: [ 1, 12, 3, 1 ] [ 1, 12, 6 ] 3: [ 1, 15 ] same dimensions for G and N_G(P) 5: [ 1, 2, 1 ] same dimensions for G and N_G(P) 17: [ 1, 8, 1 ] same dimensions for G and N_G(P) 19: [ 1, 9, 1 ] same dimensions for G and N_G(P) M24: 2: [ 1, 18, 5, 1, 1 ] [ 1, 54, 6 ] 3: [ 1, 9, 3 ] [ 1, 10, 2 ] 5: [ 1, 4 ] same dimensions for G and N_G(P) 7: [ 1, 3, 1 ] same dimensions for G and N_G(P) 11: [ 1, 10 ] same dimensions for G and N_G(P) 23: [ 1, 11, 1 ] same dimensions for G and N_G(P) McL: 2: [ 1, 12, 4, 1 ] [ 1, 16 ] 3: [ 1, 15, 4, 1 ] [ 1, 16, 4 ] 5: [ 1, 14, 4 ] [ 1, 15, 3 ] 7: [ 1, 3, 1 ] same dimensions for G and N_G(P) 11: [ 1, 5, 1 ] same dimensions for G and N_G(P) He: 2: [ 1, 18, 5, 1, 1 ] [ 1, 54, 6 ] 3: [ 1, 9, 3 ] [ 1, 10, 2 ] 5: [ 1, 14, 1 ] same dimensions for G and N_G(P) 7: [ 1, 19, 3 ] [ 1, 21, 1 ] 17: [ 1, 8, 1 ] same dimensions for G and N_G(P) Ru: 2: [ 1, 30, 1 ] [ 1, 83, 1 ] 3: [ 1, 9, 4 ] [ 1, 10, 3 ] 5: [ 1, 20, 4 ] [ 1, 21, 3 ] 7: [ 1, 6 ] same dimensions for G and N_G(P) 13: [ 1, 12 ] same dimensions for G and N_G(P) 29: [ 1, 14, 1 ] same dimensions for G and N_G(P) Suz: 2: [ 1, 26, 9, 1, 1 ] [ 1, 59, 21 ] 3: [ 1, 22, 8, 2, 1 ] [ 1, 24, 11, 3 ] 5: [ 1, 10, 5 ] same dimensions for G and N_G(P) 7: [ 1, 6 ] same dimensions for G and N_G(P) 11: [ 1, 10 ] same dimensions for G and N_G(P) 13: [ 1, 6, 1 ] same dimensions for G and N_G(P) ON: 2: [ 1, 16, 2, 1 ] [ 1, 27, 4 ] 3: [ 1, 14, 3 ] same dimensions for G and N_G(P) 5: [ 1, 4 ] same dimensions for G and N_G(P) 7: [ 1, 20, 3 ] [ 1, 21, 2 ] 11: [ 1, 10 ] same dimensions for G and N_G(P) 19: [ 1, 6, 1, 1 ] same dimensions for G and N_G(P) 31: [ 1, 15, 1 ] same dimensions for G and N_G(P) Co3: 2: [ 1, 16, 11, 3, 1 ] [ 1, 49, 11 ] 3: [ 1, 28, 8, 2 ] [ 1, 35, 15, 1 ] 5: [ 1, 20, 5 ] [ 1, 21, 4 ] 7: [ 1, 6 ] same dimensions for G and N_G(P) 11: [ 1, 5, 1 ] same dimensions for G and N_G(P) 23: [ 1, 11, 1 ] same dimensions for G and N_G(P) Co2: 2: [ 1, 36, 16, 4, 1, 1 ] [ 1, 310, 82, 1 ] 3: [ 1, 24, 16, 5, 1 ] [ 1, 30, 12, 2 ] 5: [ 1, 21, 5 ] [ 1, 22, 4 ] 7: [ 1, 6 ] same dimensions for G and N_G(P) 11: [ 1, 10 ] same dimensions for G and N_G(P) 23: [ 1, 11, 1 ] same dimensions for G and N_G(P) Fi22: 2: [ 1, 38, 14, 6, 1, 1, 1 ] [ 1, 188, 24, 2 ] 3: [ 1, 31, 17, 7, 1, 1 ] [ 1, 93, 9 ] 5: [ 1, 13, 6 ] same dimensions for G and N_G(P) 7: [ 1, 6 ] same dimensions for G and N_G(P) 11: [ 1, 5, 1 ] same dimensions for G and N_G(P) 13: [ 1, 6, 1 ] same dimensions for G and N_G(P) HN: 2: [ 1, 36, 7, 1 ] [ 1, 58, 15 ] 3: [ 1, 24, 7, 1 ] [ 1, 23, 6 ] 5: [ 1, 34, 9, 1 ] [ 1, 64, 3 ] 7: [ 1, 6 ] same dimensions for G and N_G(P) 11: [ 1, 10 ] same dimensions for G and N_G(P) 19: [ 1, 9, 1 ] same dimensions for G and N_G(P) Ly: 2: [ 1, 13, 8, 3 ] [ 1, 20, 1 ] 3: [ 1, 29, 9, 2, 1 ] [ 1, 30, 16, 4 ] 5: [ 1, 39, 7, 1 ] [ 1, 52, 5 ] 7: [ 1, 6 ] same dimensions for G and N_G(P) 11: [ 1, 5, 1 ] same dimensions for G and N_G(P) 31: [ 1, 6, 1, 1, 1, 1 ] same dimensions for G and N_G(P) 37: [ 1, 18, 1 ] same dimensions for G and N_G(P) 67: [ 1, 22, 1, 1 ] same dimensions for G and N_G(P) Th: 2: [ 1, 41, 3 ] [ 1, 88 ] 3: [ 1, 38, 2 ] [ 1, 69, 3 ] 5: [ 1, 21, 5 ] [ 1, 22, 4 ] 7: [ 1, 23, 3 ] same dimensions for G and N_G(P) 13: [ 1, 12 ] same dimensions for G and N_G(P) 19: [ 1, 18 ] same dimensions for G and N_G(P) 31: [ 1, 15, 1 ] same dimensions for G and N_G(P) Fi23: 2: [ 1, 48, 25, 7, 4, 2, 1, 1 ] [ 1, 152, 141, 23, 2 ] 3: [ 1, 51, 28, 9, 3, 1, 1 ] [ 1, 230, 58, 3 ] 5: [ 1, 13, 6 ] same dimensions for G and N_G(P) 7: [ 1, 6 ] same dimensions for G and N_G(P) 11: [ 1, 10 ] same dimensions for G and N_G(P) 13: [ 1, 6, 1 ] same dimensions for G and N_G(P) 17: [ 1, 16 ] same dimensions for G and N_G(P) 23: [ 1, 11, 1 ] same dimensions for G and N_G(P) Co1: 2: [ 1, 64, 27, 2, 1, 1 ] (no Sylow normalizer table) 3: [ 1, 36, 25, 10, 1, 1 ] [ 1, 73, 35, 19 ] 5: [ 1, 23, 14, 7 ] [ 1, 26, 9, 1 ] 7: [ 1, 12, 14 ] same dimensions for G and N_G(P) 11: [ 1, 10 ] same dimensions for G and N_G(P) 13: [ 1, 12 ] same dimensions for G and N_G(P) 23: [ 1, 11, 1 ] same dimensions for G and N_G(P) J4: 2: [ 1, 51, 6, 1 ] (no Sylow normalizer table) 3: [ 1, 9, 4 ] [ 1, 10, 3 ] 5: [ 1, 4 ] same dimensions for G and N_G(P) 7: [ 1, 3, 1 ] same dimensions for G and N_G(P) 11: [ 1, 43, 5 ] [ 1, 44, 4 ] 23: [ 1, 22 ] same dimensions for G and N_G(P) 29: [ 1, 28 ] same dimensions for G and N_G(P) 31: [ 1, 10, 1, 1 ] same dimensions for G and N_G(P) 37: [ 1, 12, 1, 1 ] same dimensions for G and N_G(P) 43: [ 1, 14, 1, 1 ] same dimensions for G and N_G(P) F3+: 2: [ 1, 71, 20, 4, 1 ] (no Sylow normalizer table) 3: [ 1, 72, 24, 4 ] (no Sylow normalizer table) 5: [ 1, 13, 6 ] same dimensions for G and N_G(P) 7: [ 1, 26, 4 ] [ 1, 27, 3 ] 11: [ 1, 10 ] same dimensions for G and N_G(P) 13: [ 1, 12 ] same dimensions for G and N_G(P) 17: [ 1, 16 ] same dimensions for G and N_G(P) 23: [ 1, 11, 1 ] same dimensions for G and N_G(P) 29: [ 1, 14, 1 ] same dimensions for G and N_G(P) B: 2: [ 1, 140, 30, 7, 1 ] (no Sylow normalizer table) 3: [ 1, 81, 41, 13, 4, 1, 1 ] [ 1, 230, 58, 3 ] 5: [ 1, 56, 18, 4 ] [ 1, 52, 5 ] 7: [ 1, 23, 3 ] same dimensions for G and N_G(P) 11: [ 1, 10 ] (no Sylow normalizer table) 13: [ 1, 12 ] (no Sylow normalizer table) 17: [ 1, 16 ] (no Sylow normalizer table) 19: [ 1, 18 ] (no Sylow normalizer table) 23: [ 1, 11, 1 ] (no Sylow normalizer table) 31: [ 1, 15, 1 ] same dimensions for G and N_G(P) 47: [ 1, 23, 1 ] (no Sylow normalizer table) M: 2: [ 1, 169, 12, 1 ] (no Sylow normalizer table) 3: [ 1, 147, 14, 2, 1 ] (no Sylow normalizer table) 5: [ 1, 106, 17, 4, 1 ] [ 1, 159, 37, 4 ] 7: [ 1, 81, 9, 1 ] [ 1, 82, 17 ] 11: [ 1, 39, 10 ] same dimensions for G and N_G(P) 13: [ 1, 56, 5 ] [ 1, 57, 4 ] 17: [ 1, 16 ] (no Sylow normalizer table) 19: [ 1, 18 ] (no Sylow normalizer table) 23: [ 1, 11, 1 ] (no Sylow normalizer table) 29: [ 1, 28 ] (no Sylow normalizer table) 31: [ 1, 15, 1 ] (no Sylow normalizer table) 41: [ 1, 40 ] (no Sylow normalizer table) 47: [ 1, 23, 1 ] same dimensions for G and N_G(P) 59: [ 1, 29, 1 ] (no Sylow normalizer table) 71: [ 1, 35, 1 ] (no Sylow normalizer table)
We see for example that for the principal block \(B\) of the group algebra of \(McL\) over a field \(F\) of characteristic \(2\), the Loewy length of \(Z(B)\) is \(4\), and the dimensions of \(J^i(Z(B))\), for \(0 \leq i \leq 4\), are \(18, 17, 5, 1, 0\), respectively. In [Sch16, p. 113], it is erroneously stated that the Loewy length is larger than \(5\), and that \(J^2(Z(B))\) has dimension \(11\).
The largest Loewy length in the above list occurs for the group \(Fi_{23}\) in characteristic \(2\).
Now let us look what we know about the abovementioned conjecture.
gap> conjecture_is_wrong_for; [ ] gap> conjecture_is_open_for; [ [ "Co1", 2 ], [ "J4", 2 ], [ "F3+", 2 ], [ "F3+", 3 ], [ "B", 2 ], [ "B", 11 ], [ "B", 13 ], [ "B", 17 ], [ "B", 19 ], [ "B", 23 ], [ "B", 47 ], [ "M", 2 ], [ "M", 3 ], [ "M", 17 ], [ "M", 19 ], [ "M", 23 ], [ "M", 29 ], [ "M", 31 ], [ "M", 41 ], [ "M", 59 ], [ "M", 71 ] ]
Most of the open cases belong to primes \(p\) that divide the group orders only once, but for which the character tables of the corresponding Sylow \(p\)-normalizers are not (yet) contained in GAP's library.
gap> defectone:= Filtered( conjecture_is_open_for, > pair -> Size( CharacterTable( pair[1] ) ) mod pair[2]^2 <> 0 ); [ [ "B", 11 ], [ "B", 13 ], [ "B", 17 ], [ "B", 19 ], [ "B", 23 ], [ "B", 47 ], [ "M", 17 ], [ "M", 19 ], [ "M", 23 ], [ "M", 29 ], [ "M", 31 ], [ "M", 41 ], [ "M", 59 ], [ "M", 71 ] ]
The following table lists the structures of the groups in question, as they are stated in [CCN+85, pp. 217, 234].
| \(G\) | \(p\) | Structure |
| \(B\) | \(11\) | \(11:10 \times S_5\) |
| \(13\) | \(13:12 \times S_4\) | |
| \(17\) | \((17:8 \times 2^2).2\) | |
| \(19\) | \(19:18 \times 2\) | |
| \(23\) | \(23:11 \times 2\) | |
| \(47\) | \(47:23\) | |
| \(M\) | \(17\) | \((17:8 \times L_3(2)).2\) |
| \(\) | \(19\) | \((19:9 \times A_5):2\) |
| \(\) | \(23\) | \(23:11 \times S_4\) |
| \(\) | \(29\) | \((29:14 \times 3).2\) |
| \(\) | \(31\) | \(31:15 \times S_3\) |
| \(\) | \(41\) | \(41:40\) |
| \(\) | \(59\) | \(59:29\) |
| \(\) | \(71\) | \(71:35\) |
We compute the dimensions of the layers in these cases.
gap> # G = B, p = 11 gap> DimensionsOfLoewyLayersByTable( > CharacterTable( "P:Q", [ 11, 10 ] ) * > CharacterTable( "S5" ), 11 ); [ 1, 10 ] gap> # G = B, p = 13 gap> DimensionsOfLoewyLayersByTable( > CharacterTable( "P:Q", [ 13, 12 ] ) * > CharacterTable( "S4" ), 13 ); [ 1, 12 ] gap> # G = B, p = 17 gap> tblH1:= CharacterTable( "17:8" );; gap> tblG1:= CharacterTable( "17:16" );; gap> tblH2:= CharacterTable( "2^2" );; gap> tblG2:= CharacterTable( "D8" );; gap> DimensionsOfLoewyLayersByTable( > CharacterTableOfIndexTwoSubdirectProduct( > tblH1, tblG1, tblH2, tblG2, "(17:8x2^2).2" ).table, 17 ); [ 1, 16 ] gap> # G = B, p = 19 gap> DimensionsOfLoewyLayersByTable( > CharacterTable( "P:Q", [ 19, 18 ] ) * > CharacterTable( "C2" ), 19 ); [ 1, 18 ] gap> # G = B, p = 23 gap> DimensionsOfLoewyLayersByTable( > CharacterTable( "P:Q", [ 23, 11 ] ) * > CharacterTable( "C2" ), 23 ); [ 1, 11, 1 ] gap> # G = B, p = 47 gap> DimensionsOfLoewyLayersByTable( > CharacterTable( "P:Q", [ 47, 23 ] ), 47 ); [ 1, 23, 1 ] gap> # G = M, p = 17 gap> tblH1:= CharacterTable( "17:8" );; gap> tblG1:= CharacterTable( "17:16" );; gap> tblH2:= CharacterTable( "L3(2)" );; gap> tblG2:= CharacterTable( "L3(2).2" );; gap> DimensionsOfLoewyLayersByTable( > CharacterTableOfIndexTwoSubdirectProduct( > tblH1, tblG1, tblH2, tblG2, "(17:8xL3(2)).2" ).table, 17 ); [ 1, 16 ] gap> # G = M, p = 19 gap> tblH1:= CharacterTable( "19:9" );; gap> tblG1:= CharacterTable( "19:18" );; gap> tblH2:= CharacterTable( "A5" );; gap> tblG2:= CharacterTable( "S5" );; gap> DimensionsOfLoewyLayersByTable( > CharacterTableOfIndexTwoSubdirectProduct( > tblH1, tblG1, tblH2, tblG2, "(19:9xA5).2" ).table, 19 ); [ 1, 18 ] gap> # G = M, p = 23 gap> DimensionsOfLoewyLayersByTable( > CharacterTable( "P:Q", [ 23, 11 ] ) * > CharacterTable( "S4" ), 23 ); [ 1, 11, 1 ] gap> # G = M, p = 29 gap> tblH1:= CharacterTable( "29:14" );; gap> tblG1:= CharacterTable( "29:28" );; gap> tblH2:= CharacterTable( "C3" );; gap> tblG2:= CharacterTable( "S3" );; gap> DimensionsOfLoewyLayersByTable( > CharacterTableOfIndexTwoSubdirectProduct( > tblH1, tblG1, tblH2, tblG2, "(29:14x3).2" ).table, 29 ); [ 1, 28 ] gap> # G = M, p = 31 gap> DimensionsOfLoewyLayersByTable( > CharacterTable( "P:Q", [ 31, 15 ] ) * > CharacterTable( "S3" ), 31 ); [ 1, 15, 1 ] gap> # G = M, p = 41 gap> DimensionsOfLoewyLayersByTable( > CharacterTable( "P:Q", [ 41, 40 ] ), 41 ); [ 1, 40 ] gap> # G = M, p = 59 gap> DimensionsOfLoewyLayersByTable( > CharacterTable( "P:Q", [ 59, 29 ] ), 59 ); [ 1, 29, 1 ] gap> # G = M, p = 71 gap> DimensionsOfLoewyLayersByTable( > CharacterTable( "P:Q", [ 71, 35 ] ), 71 ); [ 1, 35, 1 ]
For those normalizers \(N\) in the above computations that are not direct products, the subdirect product structure follows from the containment of \(N\) in known subgroups of \(B\) or \(M\), respectively:
\((17:8 \times 2^2).2 < (2^2 \times F_4(2)).2 < B\),
\((17:8 \times L_3(2)).2 < (S_4(4).2 \times L_3(2)).2 < M\),
\((19:9 \times A_5).2 < (U_3(8).3 \times A_5).2 < M\), and
\((29:14 \times 3).2 < 3.Fi_{24} < M\).
Seven cases remain to be checked. From [Wil98], we know that the Sylow \(2\)-subgroups of \(Co_1\), \(J_4\), \(F_{{3+}}\), \(B\), and \(M\) are self-normalizing in the simple group, and that the Sylow \(3\)-normalizers of \(F_{{3+}}\) and \(M\) have the \(3'\)-parts \(2^3\) and \(2^6\), respectively.
gap> hard:= Filtered( conjecture_is_open_for, > pair -> Size( CharacterTable( pair[1] ) ) mod pair[2]^2 = 0 ); [ [ "Co1", 2 ], [ "J4", 2 ], [ "F3+", 2 ], [ "F3+", 3 ], [ "B", 2 ], [ "M", 2 ], [ "M", 3 ] ]
The first four of these cases and the last one are small enough for our appproach, GAP needs several hours for each of the computations. The results are as follows.
The Sylow \(2\)-subgroup of \(Co_1\) has \(782\) conjugacy classes, the dimensions of the Loewy layers of the centre of its group algebra over the field with two elements are \(1, 766, 15\).
The Sylow \(2\)-subgroup of \(J_4\) has \(581\) conjugacy classes, the dimensions of the Loewy layers of the centre of its group algebra over the field with two elements are \(1, 553, 27\).
The Sylow \(2\)-subgroup of \(F_{3+}\) has \(581\) conjugacy classes, the dimensions of the Loewy layers of the centre of its group algebra over the field with two elements are \(1, 553, 27\).
The Sylow \(3\)-normalizer of \(F_{3+}\) is supersolvable and has \(701\) conjugacy classes, the dimensions of the Loewy layers of the centre of its group algebra over the field with three elements are \(1, 620, 78, 2\).
The Sylow \(3\)-normalizer of \(M\) has \(810\) conjugacy classes, the dimensions of the Loewy layers of the centre of its group algebra over the field with three elements are \(1, 698, 108, 3\).
The other two open cases are more involved. For example, the Sylow \(2\)-subgroup of the Monster has \(26752\) conjugacy classes. We should better think about another strategy.
Remark:
The Sylow \(2\)-subgroups of the groups \(J_4\) and \(F_{3+}\) have the same number of conjugacy classes and the same Loewy layer dimensions. In fact, they have the same irreducible characters. This implies that the centres of their group algebras are isomorphic, because the structure constants that define the multiplication in these algebras depend only on the character values, not on the element orders (see ClassMultiplicationCoefficient (Reference: ClassMultiplicationCoefficient for character tables)).
The two groups are not isomorphic, since the numbers of conjugacy classes of elements of given orders are different: The Sylow \(2\)-subgroups of \(J_4\) have \(106\), \(398\), and \(75\) conjugacy classes of elements of the orders \(2\), \(4\), and \(8\), respectively, whereas these numbers are \(98\), \(394\), and \(87\) in the case of the Sylow \(2\)-subgroups of \(F_{3+}\).
The similarity of the two Sylow \(2\)-subgroups can be explained by the fact that the groups are contained in (maximal) subgroups of the structure \(2^{11}.M_{24}\) in \(J_4\) and \(F_{3+}\), where the involved \(11\)-dimensional modules for \(M_{24}\) are isomorphic; also these maximal subgroups have the same irreducible characters. (But note that also the Sylow \(2\)-subgroup of \(Co_1\) is contained in a (maximal) subgroup of the structure \(2^{11}.M_{24}\).)
A GAP session that shows these facts is listed below. (A direct call of IsomorphismGroups (Reference: IsomorphismGroups) for the two groups did not finish after several days.) Note that the Sylow \(2\)-subgroup of \(J_4\) is contained in the maximal subgroups of the type \(2^{11}:M_{24}\), for which a permutation representation on \(2^{11}\) points is known.
First we construct the two Sylow \(2\)-subgroups, represented as PC groups (see Chapter Reference: Pc Groups).
gap> g1:= AtlasGroup( "Fi24'" );; gap> s1:= SylowSubgroup( g1, 2 );; gap> pc1:= Image( IsomorphismPcGroup( s1 ) ); gap> j4:= CharacterTable( "J4" );; gap> j4m1:= CharacterTable( Maxes( j4 )[1] );; gap> ( Size( j4 ) / Size( j4m1 ) ) mod 2; 1 gap> GroupInfoForCharacterTable( j4m1 ); [ [ "AtlasSubgroup", [ "J4", 1 ] ], [ "PrimitiveGroup", [ 2048, 11 ] ] ] gap> g2:= PrimitiveGroup( 2048, 11 );; gap> s2:= SylowSubgroup( g2, 2 );; gap> pc2:= Image( IsomorphismPcGroup( s2 ) );; gap> Collected( Factors( Size( pc1 ) ) ); [ [ 2, 21 ] ] gap> Size( pc1 ) = Size( pc2 ); true
Next we compute the two character tables and check the above claims. (Currently GAP does not recognize automatically that the PC groups are in fact \(p\)-groups, but we have to set the supersolvability flag in order to use an efficient algorithm for the computations.)
gap> IsPGroup( pc1 );; IsPGroup( pc2 );; gap> t1:= CharacterTable( pc1 );; gap> irr1:= Irr( t1 );; gap> t2:= CharacterTable( pc2 );; gap> irr2:= Irr( t2 );; gap> deg:= CharacterDegrees( t1 ); [ [ 1, 32 ], [ 2, 56 ], [ 4, 80 ], [ 8, 84 ], [ 16, 109 ], [ 32, 70 ], [ 64, 94 ], [ 128, 42 ], [ 256, 14 ] ] gap> deg = CharacterDegrees( t2 ); true gap> IsRecord( TransformingPermutations( irr1, irr2 ) ); true gap> Collected( OrdersClassRepresentatives( t1 ) ); [ [ 1, 1 ], [ 2, 98 ], [ 4, 394 ], [ 8, 87 ], [ 16, 1 ] ] gap> Collected( OrdersClassRepresentatives( t2 ) ); [ [ 1, 1 ], [ 2, 106 ], [ 4, 398 ], [ 8, 75 ], [ 16, 1 ] ]
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