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Add FrattiniSubgroup for nilpotent groups #583

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Mar 28, 2016
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Add FrattiniSubgroup for nilpotent groups #583

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cbfea49
Add FrattiniSubgroup for nilpotent groups
hungaborhorvath Feb 2, 2016
29f6920
AbelInv and IndepGens are called once at the beginning
hungaborhorvath Feb 11, 2016
2b0f3c7
Replace CommutatorFactorGroup by MaximalAbelianQuotient
hungaborhorvath Feb 11, 2016
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26 changes: 26 additions & 0 deletions lib/grp.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -1013,6 +1013,32 @@ RedispatchOnCondition( FittingSubgroup, true, [IsGroup], [IsFinite], 0);
##
#M FrattiniSubgroup( <G> ) . . . . . . . . . . Frattini subgroup of a group
##
InstallMethod( FrattiniSubgroup, "generic method for nilpotent groups",
[ IsGroup ],
RankFilter( IsGroup and IsNilpotentGroup )
- RankFilter( IsGroup ),
function(G)
local i, p, q, gen, Gf;
if IsTrivial(G) then
return G;
elif IsAbelian(G) then
gen := [ ];
for i in [1..Length(AbelianInvariants(G))] do
q := AbelianInvariants(G)[i];
if q<>0 and not IsPrime(q) then
p := SmallestRootInt(q);
Add(gen, IndependentGeneratorsOfAbelianGroup(G)[i]^p);
fi;
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Calling AbelianInvariants and IndependentGeneratorsOfAbelianGroup again and again like that makes me cringe. While we cache these values, each of these calls incurs an overhead, as the system has to perform a lookup in a component object. So please store those into temporary variables.

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@fingolfin Ok, now I call them in the beginning and then use the elements of this stored list.
The reason I did not want to do this was that if an abelian group has only invariants of 0 and prime then we would not even need to call IndependentGenerators, which could be more expensive than AbelianInvariants.

od;
return SubgroupNC(G, gen);
elif IsNilpotentGroup(G) then
Gf := CommutatorFactorGroup(G);
return PreImage(NaturalHomomorphism(Gf), FrattiniSubgroup(Gf));
else
TryNextMethod();
fi;
end);

InstallMethod( FrattiniSubgroup, "generic method for groups", [ IsGroup ],0,
function(G)
local m;
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9 changes: 4 additions & 5 deletions lib/grpperm.gi
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Expand Up @@ -1593,15 +1593,14 @@ end );
##
InstallMethod( FrattiniSubgroup,"for permgrp", true, [ IsPermGroup ], 0,
function( G )
local fac, p, l, k, i, j;
local p, l, k, i, j;

fac := Set( FactorsInt( Size( G ) ) );
if Length( fac ) > 1 then
if not IsPGroup( G ) then
TryNextMethod();
elif fac[1]=1 then
elif IsTrivial( G ) then
return G;
fi;
p := fac[ 1 ];
p := PrimePGroup( G );
l := GeneratorsOfGroup( G );
k := [ l[1]^p ];
for i in [ 2 .. Length(l) ] do
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20 changes: 20 additions & 0 deletions tst/testinstall/opers/FrattiniSubgroup.tst
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@@ -1,5 +1,25 @@
gap> START_TEST("FrattiniSubgroup.tst");

#
gap> FrattiniSubgroup(Group(()));
Group(())
gap> FrattiniSubgroup(Group((1,3),(1,2,3,4)));
Group([ (1,3)(2,4) ])
gap> D := DihedralGroup(IsFpGroup,8);;
gap> FrattiniSubgroup(D)=Group([ D.1^2 ]);
true
gap> F := FreeGroup("x", "y", "z");; x := F.1;; y := F.2;; z := F.3;;
gap> G := F/[x^(-1)*y^(-1)*x*y, x^(-1)*z^(-1)*x*z, z^(-1)*y^(-1)*z*y, x^180, y^168];;
gap> HasIsAbelian(FrattiniSubgroup(G));
true
gap> GeneratorsOfGroup(FrattiniSubgroup(G));
[ (x^-195*y^196)^6, (x^-1*y)^630, (x^-1*y)^840 ]
gap> F := FrattiniSubgroup(DirectProduct(DihedralGroup(IsFpGroup,8), SmallGroup(27,4)));;
gap> IdGroup(F);
[ 6, 2 ]
gap> HasIsNilpotentGroup(F);
true

#
gap> FrattiniSubgroup(SymmetricGroup(3));
Group(())
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