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#############################################################################
##
#W maxsub.gi Polycyc Bettina Eick
##
## Maximal subgroups of p-power index.
##
#############################################################################
##
#F MaximalSubgroupsByLayer := function( G, pcp, p )
##
## A/B is an efa layer of G. We compute the maximal subgroups of p-index
## of G which do not contain A/B.
##
BindGlobal( "MaximalSubgroupsByLayer", function( G, pcp, p )
local q, C, invs, max, inv, t, D, new;
# get characteristic
if Length( pcp ) = 0 then return []; fi;
q := RelativeOrdersOfPcp( pcp )[1];
if q <> 0 and q <> p then return []; fi;
if q = 0 then
new := List( pcp, x -> x ^ p );
new := AddIgsToIgs( new, DenominatorOfPcp( pcp ) );
new := SubgroupByIgs( G, new );
new := Pcp( GroupOfPcp( pcp ), new );
else
new := pcp;
fi;
# set up class record
C := rec( );
C.group := G;
C.super := [];
C.factor := Pcp( G, GroupOfPcp( pcp ) );
C.normal := new;
# add field
C.char := p;
C.field := GF(p);
C.dim := Length( pcp );
C.one := IdentityMat( C.dim, C.field );
# add extension info
AddRelatorsCR( C );
AddOperationCR( C );
# if it is a trivial factor
if Length( pcp ) = 1 then
AddInversesCR( C );
t := ComplementClassesCR( C );
fi;
# get maximal subgroups
invs := SMTX.BasesMaximalSubmodules(GModuleByMats(C.mats, C.dim, C.field));
# loop trough
max := [];
for inv in invs do
D := InduceToFactor( C, rec( repr := inv, stab := [] ) );
AddInversesCR( D );
t := ComplementClassesCR( D );
Append( max, t );
od;
return max;
end );
#############################################################################
##
#F MaximalSubgroupClassesByIndex( G, p )
##
## The conjugacy classes of maximal subgroups of p-power index in G.
##
InstallMethod( MaximalSubgroupClassesByIndexOp, "for pcp groups",
[IsPcpGroup, IsPosInt],
function( G, p )
local pcp, max, i, tmp;
# loop over series and determine subgroups
pcp := PcpsOfEfaSeries( G );
max := [];
for i in [1..Length(pcp)] do
Append( max, MaximalSubgroupsByLayer( G, pcp[i], p ) );
od;
# translate to classes and return
for i in [1..Length(max)] do
tmp := ConjugacyClassSubgroups( G, max[i].repr );
SetStabilizerOfExternalSet( tmp, max[i].norm );
max[i] := tmp;
od;
return max;
end );
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