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############################################################################
##
#W polyz.gi Polycyc Bettina Eick
##
############################################################################
##
#F GeneratorsOfCentralizerOfPcp( gens, pcp )
##
BindGlobal( "GeneratorsOfCentralizerOfPcp", function( gens, pcp )
local idm, v, mats;
idm := IdentityMat( Length( pcp ), GF( RelativeOrdersOfPcp(pcp)[1] ) );
for v in idm do
mats := LinearActionOnPcp( gens, pcp );
gens := PcpOrbitStabilizer( v, gens, mats, OnRight ).stab;
od;
return gens;
end );
############################################################################
##
#F PolyZNormalSubgroup( G )
##
## returns a normal subgroup N of finite index in G such that N has a
## normal series with free abelian factors.
##
InstallGlobalFunction( PolyZNormalSubgroup, function( G )
local N, F, U, ser, nat, pcps, m, i, free, j, p;
# set up
N := TrivialSubgroup( G );
ser := [N];
nat := IdentityMapping( G );
F := Image( nat );
# loop
while not IsFinite( F ) do
# get gens of free abelian normal subgroup
pcps := PcpsOfEfaSeries(F);
m := Length( pcps );
i := m;
while RelativeOrdersOfPcp( pcps[i] )[1] > 0 do
i := i - 1;
od;
free := AsList( pcps[i] );
for j in [i+1..m] do
free := GeneratorsOfCentralizerOfPcp( free, pcps[j] );
p := RelativeOrdersOfPcp( pcps[j] )[1];
if p = 2 then
free := List( free, x -> x^4 );
else
free := List( free, x -> x^p );
fi;
od;
# reset
U := Subgroup( F, free );
N := PreImage( nat, U );
Add( ser, N );
nat := NaturalHomomorphismByNormalSubgroup( G, N );
F := Image( nat );
od;
SetEfaSeries( N, Reversed( ser ) );
return N;
end );
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