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#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Frank Celler, Bettina Eick.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the methods for properties of polycyclic groups.
##
InstallMethod( IsNilpotentGroup,
"method for pc groups",
true,
[IsGroup and CanEasilyComputePcgs],
0,
function( G )
local w;
w := LGWeights( SpecialPcgs(G) );
return Last(w)[1] = 1;
end);
InstallMethod( IsSupersolvableGroup,
"method for pc groups",
true,
[IsGroup and CanEasilyComputePcgs],
0,
function( G )
local pr, spec, pcgs, p, sub, fac, mats, modu, facs;
pr := PrimeDivisors(Size(G));
spec := SpecialPcgs(G);
pcgs := InducedPcgs( spec, FrattiniSubgroup( G ) );
for p in pr do
sub := InducedPcgsByPcSequenceAndGenerators( spec, pcgs,
GeneratorsOfGroup( PCore(G, p) ) );
if Length(sub) > Length(pcgs) then
fac := sub mod pcgs;
mats := LinearOperationLayer( G, fac );
modu := GModuleByMats( mats, GF(p) );
facs := MTX.CompositionFactors( modu );
if not ForAll( facs, x -> x.dimension = 1 ) then
return false;
fi;
fi;
od;
return true;
end);
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