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#############################################################################
##
#W frattExt.gi Cubefree Heiko Dietrich
##
##
##
## These functions compute the Frattini extensions.
## Basically, already implemented methods of the GAP Package GrpConst are
## used.
##
#############################################################################
##
#F FrattiniExtensionCF( code, o )
##
## Computes the Frattini extensions of the group given by 'code' of order 'o'
InstallGlobalFunction(FrattiniExtensionCF, function( code, o )
local F, rest, primes, modus, H, i, modul, found, j, M, cc, c;
# get F and the trivial case
F := PcGroupCodeRec( code );
rest := o / code.order;
if rest = 1 then return F; fi;
# construct irreducible modules for F
primes := Factors( rest );
modus := List( primes, x -> IrreducibleModules( F, GF(x), 1 )[2] );
FindUniqueModules( modus );
# set up
H := PcGroupCodeRec( code );
# loop over primes
for i in [1..Length(primes)] do
modul := List( modus[i], x -> EnlargedModule( x, F, H ) );
found := false;
j := 0;
while not found do
j := j+1;
M := modul[j];
cc := TwoCohomology( H, M );
if Dimension( Image( cc.cohom ) ) > 0 then
c := PreImagesRepresentative( cc.cohom,
Basis(Image(cc.cohom))[1]);
H := ExtensionSQ( cc.collector, H, M, c );
found := true;
fi;
od;
od;
return H;
end);
#############################################################################
##
#F ConstructAllCFGroups( size )
##
## Computes all cube-free groups of order n up to isomorphism
##
InstallGlobalFunction(ConstructAllCFGroups, function ( size )
local cl, free, ext, t, primes, ffOrd, lv, nonAb, p, A, nSize, facNSize,
groups, arg1, arg2, pos, autPos, autGrps, tmp;
Info(InfoCF,1,"Construct all groups of order ",size,".");
# check
if not IsPosInt( size ) or not IsCubeFreeInt( size ) then
Error("Argument has to be a positive cube-free integer.");
fi;
# catch the case of size = 1
if size = 1 then
return [TrivialGroup()];
fi;
# if size is square-free then the groups of order size
# are Frattini-free and solvable
if IsSquareFreeInt(size) then
return(List(cf_FrattFreeSolvGroups(size,0,0,0)[1] ,
x->PcGroupCodeRec(x)));
fi;
# set up
groups := [];
cl := Collected( Factors( size ) );
# to store the subgroups and normalizers of GL(2,p)
autPos := [];
for t in cl do
Add(autPos,t[1]);
if t[2]=2 then
Add(autPos,t[1]^2);
fi;
od;
autGrps := ListWithIdenticalEntries( Length( autPos ), 0 );
pos := function(x) return Position( autPos, x); end;
# determine the possible non-abelian factors PSL(2,p)
nonAb:=[TrivialGroup()];
if size mod 4 = 0 then
for p in cl do
arg1 := (p[1]>3) and (size mod (p[1]*(p[1]-1)*(p[1]+1) / 2)=0);
arg2 := IsCubeFreeInt(p[1]+1) and IsCubeFreeInt(p[1]-1);
if arg1 and arg2 then
A := PSL(2,p[1]);
if Size( A )=size then
Add(groups,A );
else
Add(nonAb,A);
fi;
fi;
od;
fi;
# for every non-abelian A compute a solvable complement
for A in nonAb do
nSize := size/Size(A);
facNSize := Collected(FactorsInt(nSize));
# determine the possible Frattini-factors
primes := Product(List(facNSize,x->x[1]));
ffOrd := Filtered(DivisorsInt(nSize),x-> x mod primes =0);
free := [];
for lv in ffOrd do
tmp := cf_FrattFreeSolvGroups(lv,autPos, autGrps, pos);
autGrps := tmp[2];
free := Concatenation(free,tmp[1]);
od;
Info(InfoCF,1,"Construct ",Length(free)," Frattini extensions.");
ext := List(free,x -> FrattiniExtensionCF(x,nSize));
groups := Concatenation(groups,List(ext,x->DirectProduct(A,x)));
od;
return groups;
end );
#############################################################################
##
#F ConstructAllCFSolvableGroups( size )
##
## Computes all cube-free solvable groups of order n up to isomorphism
##
InstallGlobalFunction(ConstructAllCFSolvableGroups, function ( size )
local cl, free, ext, t, primes, ffOrd, lv, p, groups, pos, autGrps,
autPos, tmp;
# check
if not IsPosInt( size ) or not IsCubeFreeInt( size ) then
Error("Argument has to be a positive cube-free integer.");
fi;
Info(InfoCF,1,"Construct all solvable groups of order ",size,".");
# catch the case of size = 1
if size = 1 then
return [TrivialGroup()];
fi;
# if size is square-free, then the groups of order size
# are Frattini-free and solvable
if IsSquareFreeInt(size) then
return(List(cf_FrattFreeSolvGroups(size,0,0,0)[1] ,
x->PcGroupCodeRec(x)));
fi;
# set up
groups := [];
cl := Collected( Factors( size ) );
# to store the subgroups and normalizers of GL(2,p)
autPos := [];
for t in cl do
Add(autPos,t[1]);
if t[2]=2 then
Add(autPos,t[1]^2);
fi;
od;
autGrps := ListWithIdenticalEntries( Length( autPos ), 0 );
pos := function(x) return Position( autPos, x); end;
# determine the possible Frattini-factors
primes := Product(List(cl,x->x[1]));
ffOrd := Filtered(DivisorsInt(size),x-> x mod primes =0);
free := [];
for lv in ffOrd do
tmp := cf_FrattFreeSolvGroups(lv,autPos, autGrps, pos);
autGrps := tmp[2];
free := Concatenation(free,tmp[1]);
od;
Info(InfoCF,1,"Construct ",Length(free)," Frattini extensions.");
ext := List(free,x -> FrattiniExtensionCF(x,size));
groups := Concatenation(groups,ext);
return groups;
end );
##############################################################################
##
#F CubefreeTestOrder( n )
##
## Computes information about the groups of order n and compares it with the
## SmallGroups library
##
InstallGlobalFunction(CubefreeTestOrder, function( n )
local nr, nrs, ff, solv, nil, sim, all, groups, alltmp;
if not IsCubeFreeInt(n) and n<50001 and IsPosInt(n) then
Error("wrong input: need a cubefree integer between 1..50000");
fi;
all := AllSmallGroups(n);
Info(InfoCF,1,"Count groups");
nr := NumberCFGroups(n,false);
if not nr = NumberSmallGroups(n) then Error("wrong number of groups"); fi;
Info(InfoCF,1,"Count solvable groups");
nrs := NumberCFSolvableGroups(n,false);
if not nrs = Length(Filtered(all,IsSolvableGroup)) then
Error("wrong number of solvable groups");
fi;
Info(InfoCF,1,"Construct simple groups");
groups := List(ConstructAllCFSimpleGroups(n),IdSmallGroup);
alltmp := List(Filtered(all,IsSimpleGroup),IdSmallGroup);
if not Difference(groups,alltmp)=Difference(alltmp,groups) then
Error("wrong result (simple groups)");
fi;
Info(InfoCF,1,"Construct nilpotent groups");
groups := List(ConstructAllCFNilpotentGroups(n),IdSmallGroup);
alltmp := List(Filtered(all,IsNilpotentGroup),IdSmallGroup);
if not Difference(groups,alltmp)=Difference(alltmp,groups) then
Error("wrong result (nilpotent groups)");
fi;
Info(InfoCF,1,"Construct all groups");
groups := List(ConstructAllCFGroups(n),IdSmallGroup);
alltmp := List(all,IdSmallGroup);
if not Difference(groups,alltmp)=Difference(alltmp,groups) then
Error("wrong result (nilpotent groups)");
fi;
Info(InfoCF,1,"Construct Fratt.free groups");
groups := List(ConstructAllCFFrattiniFreeGroups(n),IdSmallGroup);
alltmp := List(Filtered(all,x->FrattinifactorSize(x)=n),IdSmallGroup);
if not Difference(groups,alltmp)=Difference(alltmp,groups) then
Error("wrong result (nilpotent groups)");
fi;
Info(InfoCF,1,"Everything ok");
return true;
end);
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