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Add MinimalNormalSubgroups and Socle for all nilpotent groups #606

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Mar 28, 2016
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Add MinimalNormalSubgroups and Socle for all nilpotent groups #606

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162 changes: 137 additions & 25 deletions lib/grp.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -1626,43 +1626,87 @@ end );

#############################################################################
##
#M Socle( <G> ) . . . . . . . . . . . . . . . . for finite nilpotent groups
#M Socle( <G> ) . . . . . . . . . . . . . . . . . . . . . for simple groups
##
InstallMethod( Socle, "for finite nilpotent groups",
InstallMethod( Socle, "for simple groups",
[ IsGroup and IsSimpleGroup ], SUM_FLAGS, IdFunc );

#############################################################################
##
#M Socle( <G> ) . . . . . . . . . . . . . . . for elementary abelian groups
##
InstallMethod( Socle, "for elementary abelian groups",
[ IsGroup and IsElementaryAbelian ], SUM_FLAGS, IdFunc );

#############################################################################
##
#M Socle( <G> ) . . . . . . . . . . . . . . . . . . . . for nilpotent groups
##
InstallMethod( Socle, "for nilpotent groups",
[ IsGroup ],
RankFilter( IsGroup and CanComputeSize and IsFinite and
IsNilpotentGroup ) - RankFilter( IsGroup ),
RankFilter( IsGroup and IsNilpotentGroup and IsFinite )
- RankFilter( IsGroup ),
function(G)
local H, prodH;
local P, C, size, gen, abinv, indgen, i, p, q, soc;

if not CanComputeSize(G) or not IsFinite(G)
or not IsNilpotentGroup(G) then
# IsNilpotent check might error for fp-groups
if not IsAbelian(G) and not IsNilpotentGroup(G) then
TryNextMethod();
fi;

prodH := TrivialSubgroup(G);
# now socle is the product of Omega of the Sylow subgroups of the center
for H in SylowSystem(Center(G)) do
prodH := ClosureSubgroupNC(prodH, Omega(H, PrimePGroup(H)));
od;
# for finite groups the usual methods are faster
# for SylowSystem and Omega
if ( CanComputeSize(G) or HasIsFinite(G) ) and IsFinite(G) then
soc := TrivialSubgroup(G);
# now socle is the product of Omega of Sylow subgroups of the center
for P in SylowSystem(Center(G)) do
soc := ClosureSubgroupNC(soc, Omega(P, PrimePGroup(P)));
od;
else
# compute generators for the torsion Omega p-subgroups of the center
C := Center(G);
gen := [ ];
abinv := [ ];
indgen := [ ];
size := 1;
for i in [1..Length(AbelianInvariants(C))] do
q := AbelianInvariants(C)[i];
if q<>0 then
p := SmallestRootInt(q);
if not IsBound(gen[p]) then
gen[p] := [ IndependentGeneratorsOfAbelianGroup(C)[i]^(q/p) ];
else
Add(gen[p], IndependentGeneratorsOfAbelianGroup(C)[i]^(q/p));
fi;
size := size * p;
Add(abinv, p);
Add(indgen, IndependentGeneratorsOfAbelianGroup(C)[i]^(q/p));
fi;
od;
# Socle is the product of the torsion Omega p-groups of the center
soc := Subgroup(G, Concatenation(Compacted(gen)));
SetSize(soc, size);
SetAbelianInvariants(soc, abinv);
SetIndependentGeneratorsOfAbelianGroup(soc, indgen);
fi;

# Socle is central in G, set some properties and attributes accordingly
SetIsAbelian(prodH, true);
if not HasParent(prodH) then
SetParent(prodH, G);
SetCentralizerInParent(prodH, G);
SetIsNormalInParent(prodH, true);
elif CanComputeIsSubset(G, Parent(prodH))
and IsSubgroup(G, Parent(prodH)) then
SetCentralizerInParent(prodH, Parent(prodH));
SetIsNormalInParent(prodH, true);
elif CanComputeIsSubset(G, Parent(prodH))
and IsSubgroup(Parent(prodH), G) and IsNormal(Parent(prodH), G) then
SetIsAbelian(soc, true);
if not HasParent(soc) then
SetParent(soc, G);
SetCentralizerInParent(soc, G);
SetIsNormalInParent(soc, true);
elif CanComputeIsSubset(G, Parent(soc))
and IsSubgroup(G, Parent(soc)) then
SetCentralizerInParent(soc, Parent(soc));
SetIsNormalInParent(soc, true);
elif CanComputeIsSubset(G, Parent(soc))
and IsSubgroup(Parent(soc), G) and IsNormal(Parent(soc), G) then
# characteristic subgroup of a normal subgroup is normal
SetIsNormalInParent(prodH, true);
SetIsNormalInParent(soc, true);
fi;

return prodH;
return soc;
end);

#############################################################################
Expand Down Expand Up @@ -4323,6 +4367,74 @@ InstallMethod (MinimalNormalSubgroups,
end);


#############################################################################
##
#M MinimalNormalSubgroups( <G> )
##
InstallMethod( MinimalNormalSubgroups, "for simple groups",
[ IsGroup and IsSimpleGroup ], SUM_FLAGS,
function(G) return [ G ]; end);


#############################################################################
##
#M MinimalNormalSubgroups (<G>)
##
InstallMethod( MinimalNormalSubgroups, "for nilpotent groups",
[ IsGroup ], RankFilter( IsGroup and IsFinite and IsNilpotentGroup ) - RankFilter( IsGroup ),
function(G)
local soc, i, p, primes, gen, min, MinimalSubgroupsOfPGroupByGenerators;

# IsNilpotent check might error for fp-groups
if not IsAbelian(G) and not IsNilpotentGroup(G) then
TryNextMethod();
fi;

MinimalSubgroupsOfPGroupByGenerators := function(G, p, gen)
# G is the big group
# p is the prime p
# gens is the generators by which the p-group is given
local min, tuples, g, h, k;

min := [ ];
if Length(gen[p])=1 then
Add(min, Subgroup(G, gen[p]));
else
g := Remove(gen[p]);
for tuples in IteratorOfTuples([0..p-1], Length(gen[p])) do
h := g;
for i in [1..Length(tuples)] do
h := h*gen[p][i]^tuples[i];
od;
Add(min, Subgroup(G, [h]));
od;
Append(min, MinimalSubgroupsOfPGroupByGenerators(G, p, gen));
fi;

return min;
end;

soc := Socle(G);
primes := [ ];
gen := [ ];
min := [ ];
for i in [1..Length(AbelianInvariants(soc))] do
p := AbelianInvariants(soc)[i];
AddSet(primes, p);
if not IsBound(gen[p]) then
gen[p] := [ IndependentGeneratorsOfAbelianGroup(soc)[i] ];
else
Add(gen[p], IndependentGeneratorsOfAbelianGroup(soc)[i]);
fi;
od;

for p in primes do
Append(min, MinimalSubgroupsOfPGroupByGenerators(G, p, gen));
od;
return min;
end);


#############################################################################
##
#M SmallGeneratingSet(<G>)
Expand Down
126 changes: 126 additions & 0 deletions tst/testinstall/opers/MinimalNormalSubgroups.tst
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@@ -0,0 +1,126 @@
gap> START_TEST("Socle.tst");
gap> MinimalNormalSubgroups(Group(()));
[ ]
gap> G := Group(());; NormalSubgroups(G);; MinimalNormalSubgroups(G);
[ ]
gap> D := DihedralGroup(8);;
gap> MinimalNormalSubgroups(D) = [ Center(D) ];
true
gap> List(MinimalNormalSubgroups(D), IdGroup);
[ [ 2, 1 ] ]
gap> D := DihedralGroup(IsFpGroup, 8);;
gap> MinimalNormalSubgroups(D) = [ Center(D) ];
true
gap> List(MinimalNormalSubgroups(D), IdGroup);
[ [ 2, 1 ] ]
gap> D := DihedralGroup(IsPcpGroup, 8);;
gap> MinimalNormalSubgroups(D) = [ Center(D) ];
true
gap> List(MinimalNormalSubgroups(D), IdGroup);
[ [ 2, 1 ] ]
gap> D := Group((1,3),(1,2,3,4));;
gap> MinimalNormalSubgroups(D) = [ Center(D) ];
true
gap> List(MinimalNormalSubgroups(D), IdGroup);
[ [ 2, 1 ] ]
gap> DDD := DirectProduct(D, D, D);;
gap> List(MinimalNormalSubgroups(DDD), IdGroup);
[ [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ] ]
gap> Q := QuaternionGroup(8);;
gap> MinimalNormalSubgroups(Q) = [ Center(Q) ];
true
gap> List(MinimalNormalSubgroups(Q), IdGroup);
[ [ 2, 1 ] ]
gap> MinimalNormalSubgroups(SymmetricGroup(4)) = [ Group((1,2)(3,4),(1,3)(2,4)) ];
true
gap> List(MinimalNormalSubgroups(SymmetricGroup(5)), IdGroup);
[ [ 60, 5 ] ]
gap> List(MinimalNormalSubgroups(AlternatingGroup(5)), IdGroup);
[ [ 60, 5 ] ]
gap> G := Group((1,2),(3,4),(5,6),(7,8));;
gap> IsElementaryAbelian(G);
true
gap> Size(MinimalNormalSubgroups(G));
15
gap> MinimalNormalSubgroups(PrimitiveGroup(8,3)) = [ Group([ (1,7)(2,8)(3,5)(4,6), (1,3)(2,4)(5,7)(6,8), (1,2)(3,4)(5,6)(7,8) ]) ];
true
gap> k := 5;; P := SylowSubgroup(SymmetricGroup(4*k), 2);; A := Group((4*k+1, 4*k+2, 4*k+3));; G := ClosureGroup(P, A);;
gap> Set(MinimalNormalSubgroups(G)) = Set([ Group([ (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) ]), Group([ (17,18)(19,20) ]), Group([ (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20) ]), Group([ (21,22,23) ]) ]);
true
gap> A := DihedralGroup(16);;
gap> B := SmallGroup(27, 3);;
gap> C := SmallGroup(125, 4);;
gap> D := DirectProduct(A, B, C, SmallGroup(1536, 2));;
gap> List(MinimalNormalSubgroups(D), IdGroup);
[ [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ],
[ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ],
[ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ],
[ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 2, 1 ],
[ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 3, 1 ], [ 3, 1 ], [ 3, 1 ], [ 3, 1 ],
[ 5, 1 ] ]
gap> G := Group([ (4,8)(6,10), (4,6,10,8,12), (2,4,12)(6,10,8), (3,9)(4,6,10,8,12)(7,11), (3,5)(4,6,10,8,12)(9,11), (1,3,11,9,5)(4,6,10,8,12) ]);;
gap> MinimalNormalSubgroups(G) = Set([ Group([ (6,12)(8,10), (2,10)(4,12), (2,12)(6,10) ]), Group([ (5,11)(7,9), (3,9)(7,11), (1,9,5,11,7) ]) ]);
true
gap> F := AbelianPcpGroup([0,0,0]);;
gap> G := F / Subgroup(F, [(F.1*F.2)^180, (F.1*F.2^5)^168]);;
gap> MinimalNormalSubgroups(G);
[ Pcp-group with orders [ 2 ], Pcp-group with orders [ 2 ],
Pcp-group with orders [ 2 ], Pcp-group with orders [ 3 ],
Pcp-group with orders [ 3 ], Pcp-group with orders [ 3 ],
Pcp-group with orders [ 3 ], Pcp-group with orders [ 5 ],
Pcp-group with orders [ 7 ] ]
gap> F := FreeGroup("x", "y", "z");;
gap> x := F.1;; y := F.2;; z := F.3;;
gap> F := FreeGroup("x", "y", "z");;
gap> 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*y)^180, (x*y^5)^168];;
gap> Size(MinimalNormalSubgroups(G));
9
gap> MinimalNormalSubgroups(HeisenbergPcpGroup(3));
[ ]
gap> for G in AllGroups(60) do NormalSubgroups(G);; Print(Collected(List(Set(MinimalNormalSubgroups(G)), IdGroup)), "\n"); od;
[ [ [ 2, 1 ], 1 ], [ [ 3, 1 ], 1 ], [ [ 5, 1 ], 1 ] ]
[ [ [ 2, 1 ], 1 ], [ [ 3, 1 ], 1 ], [ [ 5, 1 ], 1 ] ]
[ [ [ 2, 1 ], 1 ], [ [ 3, 1 ], 1 ], [ [ 5, 1 ], 1 ] ]
[ [ [ 2, 1 ], 1 ], [ [ 3, 1 ], 1 ], [ [ 5, 1 ], 1 ] ]
[ [ [ 60, 5 ], 1 ] ]
[ [ [ 3, 1 ], 1 ], [ [ 5, 1 ], 1 ] ]
[ [ [ 3, 1 ], 1 ], [ [ 5, 1 ], 1 ] ]
[ [ [ 3, 1 ], 1 ], [ [ 5, 1 ], 1 ] ]
[ [ [ 4, 2 ], 1 ], [ [ 5, 1 ], 1 ] ]
[ [ [ 2, 1 ], 1 ], [ [ 3, 1 ], 1 ], [ [ 5, 1 ], 1 ] ]
[ [ [ 2, 1 ], 1 ], [ [ 3, 1 ], 1 ], [ [ 5, 1 ], 1 ] ]
[ [ [ 2, 1 ], 1 ], [ [ 3, 1 ], 1 ], [ [ 5, 1 ], 1 ] ]
[ [ [ 2, 1 ], 3 ], [ [ 3, 1 ], 1 ], [ [ 5, 1 ], 1 ] ]
gap> G := SmallGroup(120,5);; List(MinimalNormalSubgroups(G), IdGroup);
[ [ 2, 1 ] ]
gap> G := SmallGroup(120,34);; List(MinimalNormalSubgroups(G), IdGroup);
[ [ 60, 5 ] ]
gap> G := SmallGroup(120,35);; List(MinimalNormalSubgroups(G), IdGroup);
[ [ 2, 1 ], [ 60, 5 ] ]
gap> for G in AllGroups(240) do if not IsSolvable(G) then NormalSubgroups(G);; Print(List(MinimalNormalSubgroups(G), IdGroup), "\n"); fi; od;
[ [ 2, 1 ] ]
[ [ 2, 1 ] ]
[ [ 2, 1 ], [ 60, 5 ] ]
[ [ 2, 1 ], [ 60, 5 ] ]
[ [ 2, 1 ] ]
[ [ 2, 1 ], [ 2, 1 ], [ 2, 1 ] ]
[ [ 2, 1 ], [ 60, 5 ] ]
[ [ 2, 1 ], [ 2, 1 ], [ 2, 1 ], [ 60, 5 ] ]
gap> for G in AllGroups(360) do if not IsSolvable(G) then NormalSubgroups(G);; Print(List(MinimalNormalSubgroups(G), IdGroup), "\n"); fi; od;
[ [ 2, 1 ], [ 3, 1 ] ]
[ [ 360, 118 ] ]
[ [ 3, 1 ], [ 60, 5 ] ]
[ [ 3, 1 ], [ 60, 5 ] ]
[ [ 3, 1 ], [ 60, 5 ] ]
[ [ 2, 1 ], [ 3, 1 ], [ 60, 5 ] ]
gap> G := AbelianGroup([2, 3, 4, 5, 6, 7, 8, 9, 10]);;
gap> Collected(List(Set(MinimalNormalSubgroups(G)), Size));
[ [ 2, 31 ], [ 3, 13 ], [ 5, 6 ], [ 7, 1 ] ]
gap> G := ElementaryAbelianGroup(2^10);;
gap> Collected(List(Set(MinimalNormalSubgroups(G)), Size));
[ [ 2, 1023 ] ]
gap> G := ElementaryAbelianGroup(7^4);;
gap> Collected(List(Set(MinimalNormalSubgroups(G)), Size));
[ [ 7, 400 ] ]
gap> STOP_TEST("Socle.tst", 10000);
36 changes: 36 additions & 0 deletions tst/testinstall/opers/Socle.tst
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Original file line number Diff line number Diff line change
@@ -1,11 +1,27 @@
gap> START_TEST("Socle.tst");
gap> Socle(Group(()));
Group(())
gap> D := DihedralGroup(8);;
gap> Socle(D) = ClosureSubgroup(TrivialSubgroup(D), Union(Set(MinimalNormalSubgroups(D), GeneratorsOfGroup)));
true
gap> IdGroup(Socle(D));
[ 2, 1 ]
gap> Socle(D) = Center(D);
true
gap> D := DihedralGroup(IsFpGroup, 8);;
gap> Socle(D) = ClosureSubgroup(TrivialSubgroup(D), Union(Set(MinimalNormalSubgroups(D), GeneratorsOfGroup)));
true
gap> IdGroup(Socle(D));
[ 2, 1 ]
gap> Socle(D) = Center(D);
true
gap> D := DihedralGroup(IsPcpGroup, 8);;
gap> Socle(D) = ClosureSubgroup(TrivialSubgroup(D), Union(Set(MinimalNormalSubgroups(D), GeneratorsOfGroup)));
true
gap> IdGroup(Socle(D));
[ 2, 1 ]
gap> Socle(D) = Center(D);
true
gap> D := Group((1,3),(1,2,3,4));;
gap> Socle(D) = ClosureSubgroup(TrivialSubgroup(D), Union(Set(MinimalNormalSubgroups(D), GeneratorsOfGroup)));
true
Expand All @@ -31,6 +47,13 @@ gap> Socle(SymmetricGroup(4)) = Group((1,2)(3,4),(1,3)(2,4));
true
gap> IdGroup(Socle(SymmetricGroup(5)));
[ 60, 5 ]
gap> IdGroup(Socle(AlternatingGroup(5)));
[ 60, 5 ]
gap> G := Group((1,2),(3,4),(5,6),(7,8));;
gap> IsElementaryAbelian(G);
true
gap> Socle(G)=G;
true
gap> Socle(PrimitiveGroup(8,3)) = Group([ (1,7)(2,8)(3,5)(4,6), (1,3)(2,4)(5,7)(6,8), (1,2)(3,4)(5,6)(7,8) ]);
true
gap> k := 5;; P := SylowSubgroup(SymmetricGroup(4*k), 2);; A := Group((4*k+1, 4*k+2, 4*k+3));; G := ClosureGroup(P, A);;
Expand All @@ -44,8 +67,21 @@ gap> IdGroup(Socle(D));
[ 1440, 5958 ]
gap> Socle(D) = Center(D);
true
gap> Socle(FittingSubgroup(SymmetricGroup(4)));
Group([ (1,4)(2,3), (1,3)(2,4) ])
gap> G := Group([ (4,8)(6,10), (4,6,10,8,12), (2,4,12)(6,10,8), (3,9)(4,6,10,8,12)
> (7,11), (3,5)(4,6,10,8,12)(9,11), (1,3,11,9,5)(4,6,10,8,12) ]);;
gap> Socle(G) = Group([ (3,7)(5,9), (5,11)(7,9), (1,5,3)(7,11,9), (2,8,10)(4,6,12), (4,6)(10,12) ]);
true
gap> F := AbelianPcpGroup([0,0,0]);;
gap> G := F / Subgroup(F, [(F.1*F.2)^180, (F.1*F.2^5)^168]);;
gap> Socle(G);
Pcp-group with orders [ 6, 210 ]
gap> F := FreeGroup("x", "y", "z");;
gap> 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*y)^180, (x*y^5)^168];;
gap> Size(Socle(G));
1260
gap> Socle(HeisenbergPcpGroup(3));
Pcp-group with orders [ ]
gap> STOP_TEST("Socle.tst", 10000);