[GAP Forum] Not G-conjugate subgroups

[Stefan Kohl]

Dear Forum,

Vahid Dabbaghian-Abdoly asked:

> If P is a Sylow subgroup of G, how can I get all not G-conjugate subgroups
> of P without computing the lattice of subgroups of G?

[ Once again -- in my last mail an overly long line has been cut, which
  I now have split into several shorter lines. Apologies. ]

Of course there is no need to compute the whole subgroup lattice of G --
you can just compute representatives for the conjugacy classes of subgroups
of P, and then get the desired result by testing conjugacy in G.

Taking a Sylow 2 subgroup of M_11 as an easy example we can illustrate this
as follows:

gap> G := MathieuGroup(11);
Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ])
gap> P := SylowSubgroup(G,2);
Group([ (2,6)(3,5)(4,8)(7,11), (3,7)(4,8)(5,11)(9,10),
  (2,3,4,7,6,5,8,11)(9,10) ])
gap> subP := List(ConjugacyClassesSubgroups(P),Representative);
[ Group(()), Group([ (2,6)(3,5)(4,8)(7,11) ]),
  Group([ (3,7)(4,8)(5,11)(9,10) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (3,7)(4,8)(5,11)(9,10) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (2,4,6,8)(3,7,5,11) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (2,3,6,5)(4,11,8,7) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (2,8)(4,6)(7,11)(9,10),
      (3,7)(4,8)(5,11)(9,10) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (2,3,4,7,6,5,8,11)(9,10),
      (2,4,6,8)(3,7,5,11) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (2,7,6,11)(3,8,5,4), (2,4,6,8)(3,7,5,11) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (2,8)(4,6)(7,11)(9,10),
      (3,7)(4,8)(5,11)(9,10), (2,3,4,7,6,5,8,11)(9,10) ]) ]
gap> Length(subP);
10
gap> reps := [subP[1]];;
gap> repeat
>      H := First(subP,grp->not ForAny(reps,rep->IsConjugate(G,grp,rep)));
>      if H <> fail then Add(reps,H); fi;
>    until H = fail;
gap> reps; # The desired set of representatives.
[ Group(()), Group([ (2,6)(3,5)(4,8)(7,11) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (3,7)(4,8)(5,11)(9,10) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (2,4,6,8)(3,7,5,11) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (2,8)(4,6)(7,11)(9,10),
      (3,7)(4,8)(5,11)(9,10) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (2,3,4,7,6,5,8,11)(9,10),
      (2,4,6,8)(3,7,5,11) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (2,7,6,11)(3,8,5,4), (2,4,6,8)(3,7,5,11) ]),
  Group([ (2,6)(3,5)(4,8)(7,11), (2,8)(4,6)(7,11)(9,10),
      (3,7)(4,8)(5,11)(9,10), (2,3,4,7,6,5,8,11)(9,10) ]) ]
gap> Length(reps);
8

Hope this helps,

    Stefan Kohl