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LCSFactorTypes := function(G)
local l, r, i, a;
l := LowerCentralSeries(G);
r := [];
for i in [1..Length(l)-1] do
a := AbelianInvariants(l[i]/l[i+1]);
Add( r, a );
od;
return r;
end;
LCSFactorSizes := function(G)
local l, r, i;
l := LowerCentralSeries(G);
r := [];
for i in [1..Length(l)-1] do
Add( r, Index(l[i],l[i+1]) );
od;
return r;
end;
WidthPGroup := function(G)
local l;
if not IsPGroup(G) then return fail; fi;
l := LCSFactorSizes(G);
l := List(l, x -> Length(Factors(x)));
return Maximum(l);
end;
SubgroupRank := function(G)
local grps;
if not IsPGroup(G) then return fail; fi;
grps := ConjugacyClassesSubgroups(G);
grps := List(grps, Representative);
grps := List(grps, RankPGroup);
return Maximum(grps);
end;
MaxNormals := function(G,N)
local C, h, m;
C := CommutatorSubgroup(G,N);
h := NaturalHomomorphismByNormalSubgroup(N,C);
m := MaximalSubgroupClassReps(Image(h));
return List(m, x -> PreImage(h,x));
end;
Obliquity := function(G)
local l, M, m, o, i, c, U, max;
if not IsPGroup(G) then return fail; fi;
# set up
l := LowerCentralSeries(G);
M := []; M[1] := [];
m := []; m[1] := G;
o := []; o[1] := 0;
# loop
for i in [2..Length(l)-1] do
# M[i]
M[i] := [l[i-1]];
c := 0;
while c < Length(M[i]) do
c := c+1;
U := M[i][c];
max := MaxNormals(G,U);
max := Filtered(max, x -> not IsSubgroup(l[i],x));
max := Filtered(max, x -> not x in M[i]);
Append(M[i], max);
od;
# m[i]
m[i] := Intersection(m[i-1], l[i]);
for U in M[i] do
m[i] := Intersection(m[i], U);
od;
# o[i]
o[i] := Index(l[i], m[i]);
if o[i] = 1 then
o[i] := 0;
else
o[i] := Length(Factors(o[i]));
fi;
od;
return Maximum(o);
end;
HasObliquityZero := function(G)
local l, M, m, i, c, U, max;
if not IsPGroup(G) then return fail; fi;
# set up
l := LowerCentralSeries(G);
M := []; M[1] := [];
m := []; m[1] := G;
# loop
for i in [2..Length(l)-1] do
# Mi
M[i] := [l[i-1]];
c := 0;
while c < Length(M[i]) do
c := c+1;
U := M[i][c];
max := MaxNormals(G,U);
max := Filtered(max, x -> not IsSubgroup(l[i],x));
max := Filtered(max, x -> not x in M[i]);
Append(M[i], max);
od;
# m[i]
m[i] := Intersection(m[i-1], l[i]);
for U in M[i] do
m[i] := Intersection(m[i], U);
if Index(l[i], m[i]) > 1 then return false; fi;
od;
od;
return true;
end;
HasObliquityZero2 := function( arg )
local g, i, N, c, low, n;
if Length( arg ) = 0 or Length( arg ) > 2 then
Error( "Syntax: HasObliquityZero( g [, i])\n");
fi;
g := arg[1];
if Length( arg ) = 2 then
i := arg[2];
else
i := 1;
fi;
low := LowerCentralSeries( g );
N := Filtered( NormalSubgroups( g ), n -> not IsSubset( n, low[i] ) );
for n in N do
c := NilpotencyClassOfGroup( g / n );
if not IsSubset( low[c], n ) then
return false;
fi;
od;
return true;
end;
HasRank3 := function(G)
return SubgroupRank(G) = 3;
end;
[ Dauer der Verarbeitung: 0.25 Sekunden
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