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#############################################################################
##
#W irred.gi GrpConst Bettina Eick
#W Hans Ulrich Besche
##
#############################################################################
##
#F Check if a module is faithful
##
InstallGlobalFunction( IsFaithfulModule, function( L, m )
local pcgs, M, iso, s;
# set up and trivial case
pcgs := Pcgs(L);
if Length( pcgs ) = 0 then
return true;
fi;
# otherwise use a zassenhaus-like method
M := Group( m.generators );
iso := GroupHomomorphismByImagesNC( L, M, pcgs, m.generators );
s := Size( KernelOfMultiplicativeGeneralMapping(iso) );
return s = 1;
end );
#############################################################################
##
#F Check size of Sylow case
##
InstallGlobalFunction( IsMaximalPrimePower, function( a, b )
local fb, fa;
fb := Factors(b);
fa := Factors(a/b);
return b <> 1 and Length(Set(fb))=1 and not fb[1] in fa;
end );
#############################################################################
##
#F GModuleByGroup( G )
##
InstallGlobalFunction( GModuleByGroup, function( G )
return GModuleByMats( GeneratorsOfGroup(G),
DimensionOfMatrixGroup(G),
FieldOfMatrixGroup(G) );
end );
#############################################################################
##
#F Check conjugacy of groups
##
BindGlobal( "AreConjugateGroups", function( M, G, H )
local nat, g, iso, aut, a, h, C, i, r;
Print(" compute images \n");
nat := IsomorphismPermGroup( M );
G := Image( nat, G );
H := Image( nat, H );
M := Image( nat, M );
Print(" check equality \n");
if G = H then return true; fi;
g := GeneratorsOfGroup(G);
iso := IsomorphismGroups( G, H );
aut := AutomorphismGroup( G );
Print(" automorphism group has size ",Size(aut),"\n");
for a in Elements(aut) do
Print(" try next automorphism \n");
h := List( g, x -> Image( iso, Image(a, x) ) );
C := M;
r := ();
i := 1;
while i <= Length( g ) and not IsBool(r) do
Print(" ",i,"th generator \n");
r := RepresentativeAction( C, g[i], h[i] );
if not IsBool( r ) then
C := Centralizer( C, g[i] );
r := r^-1;
h := List( h, x -> x^r );
fi;
i := i+1;
od;
if not IsBool(r) then return true; fi;
od;
return false;
end );
InstallGlobalFunction( ReduceToClasses, function( M, list )
local gens, reps, U, orb, c, g, new;
# the trivial case
if Length( list ) = 0 or Length( list ) = 1 then return list; fi;
if Length( list ) = 2 then
if AreConjugateGroups( M, list[1], list[2] ) then
return [list[1]];
else
return list;
fi;
fi;
# start to work
gens := GeneratorsOfGroup( M );
reps := [];
while Length( list ) > 0 do
U := list[Length(list)];
list := Filtered( list, x -> x <> U );
Add( reps, U );
orb := [U];
c := 1;
while c <= Length( orb ) and Length(list) > 0 do
for g in gens do
new := orb[c]^g;
if not new in orb then
list := Filtered( list, x -> x <> new );
Add( orb, new );
fi;
od;
c := c + 1;
od;
od;
return reps;
end );
#############################################################################
##
#F Let L be a soluble group. Find all irreducible embeddings of L into
## GL(n,p) up to conjugacy.
##
InstallGlobalFunction( IrreducibleEmbeddings, function( n, p, L )
local M, f, modu, cl, m, U;
# we don't want to see the trivial case here
if n = 1 then return false; fi;
# first check the arguments and avoid trivial cases
if Size( PCore( L, p ) ) > 1 then return []; fi;
# compute modules
f := GF(p);
modu := IrreducibleModules( L, f, n )[2];
modu := Filtered( modu, x -> x.dimension = n );
# filter out non-faithful ones
modu := Filtered( modu, x -> IsFaithfulModule(L,x));
# reduce to conjugacy classes
M := GL(n,p);
cl := [];
for m in modu do
U := SubgroupNC( M, m.generators );
SetSize(U, Size(L));
Add( cl, U );
od;
cl := ReduceToClasses( M, cl );
return cl;
end );
#############################################################################
##
#F Compute all irreducible soluble subgroups of GL(n,p) of order dividing
## size up to conjugacy. Consider a number of different cases.
##
#############################################################################
InstallGlobalFunction( IrreducibleGroupsByAbelian, function(n, p, size)
local P, primes, cl, q, new, M, field, root, iso, i, tmp;
P := CyclicGroup( IsPermGroup, p - 1 );
primes := Set(FactorsInt(Size(P)));
primes := Filtered( primes, x -> x <> 1 );
cl := [TrivialSubgroup(P)];
for q in primes do
new := Pcgs( SylowSubgroup(P, q) );
new := List(cl, x -> List( new, y -> ClosureGroup( x, y )));
cl := Concatenation( cl, Flat( new ) );
cl := Filtered( cl, x -> IsInt( size / Size(x) ) );
od;
# compute isomorphism
M := GL(n,p);
field := GF( p );
root := PrimitiveRoot( field );
iso := GroupHomomorphismByImagesNC( P, M,
GeneratorsOfGroup( P ), [[[root]]] );
# convert
for i in [1..Length(cl)] do
tmp := Image( iso, cl[i] );
SetSize( tmp, Size( cl[i] ) );
cl[i] := tmp;
od;
return cl;
end );
#############################################################################
InstallGlobalFunction( IrreducibleGroupsByCatalogue, function(n,p,size)
local k, cl;
# we don't want to see the trivial case here
if n = 1 then return false; fi;
# return list from irredsol
return AllIrreducibleSolvableMatrixGroups( Degree, [n], Field, [GF(p)],
Order, DivisorsInt(size) );
# get irreducible groups - old version based on primitive groups
k := NumberIrreducibleSolvableGroups( n, p );
cl := List( [1..k], x -> IrreducibleSolvableGroupMS( n, p, x ) );
cl := Filtered( cl, x -> IsInt( size / Size(x) ) );
return cl;
end );
#############################################################################
InstallGlobalFunction( IrreducibleGroupsByEmbeddings, function(n,p,size)
local div, all, cl, L, M, d, q, P, m;
# we don't want to see the trivial case here
if n = 1 then return false; fi;
# create the possible isomorphism types of groups
M := GL(n,p);
div := DivisorsInt( size );
div := Filtered( div, x -> x <> 1 and IsInt(Size(M)/x) );
cl := [];
for d in div do
# first the very special case that we have the full size
if d = Size(M) then
if n=2 and (p=2 or p=3) then Add( cl, M ); fi;
# now consider the Sylow case - extend to p-group case?
elif IsMaximalPrimePower( Size(M), d ) then
q := Factors(d)[1];
if q <> p then
P := SylowSubgroup(M, q );
m := GModuleByGroup( P );
if MTX.IsIrreducible( m ) then Add( cl, P ); fi;
fi;
# in all other cases loop over all groups
else
all := AllSmallGroups( d, IsSolvableGroup );
for L in all do
Append( cl, IrreducibleEmbeddings( n, p, L ) );
od;
fi;
od;
return cl;
end );
#############################################################################
InstallGlobalFunction( IrreducibleGroups, function( n, p, size )
if n = 1 then
return IrreducibleGroupsByAbelian(n,p,size);
#elif p^n < 256 and PRIM_AVAILABLE then
elif IsAvailableIrreducibleSolvableGroupData(n,p) then
return IrreducibleGroupsByCatalogue(n,p,size);
elif SmallGroupsAvailable(size) then
return IrreducibleGroupsByEmbeddings(n,p,size);
else
return fail;
fi;
end );
#############################################################################
##
#F Test to check embeddings versus catalogue
##
BindGlobal( "TestIrred", function(limit, start)
local ppi, q, p, n, max, size, emb, cat;
ppi := Filtered([start..255], x -> IsPrimePowerInt(x) and not IsPrime(x));
for q in ppi do
p := Factors(q)[1];
n := Length( Factors(q) );
max := QuoInt( limit, q );
for size in [1..max] do
Print("start ", p, "^", n," and size ", size,"\n");
emb := IrreducibleGroupsByEmbeddings( n, p, size );
emb := List( emb, IdGroup );
Sort( emb );
cat := IrreducibleGroupsByCatalogue( n, p, size );
cat := List( cat, IdGroup );
Sort( cat );
if cat <> emb then
Error("hier\n");
fi;
od;
od;
end );
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2026-03-28
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