[GAP Forum] Contain a fixed subgroup

[William DeMeo]

Dear Mr. Sorouhesh,

I believe the command IntermediateSubgroups does what you want.
Here's an example (where I'm calling your "fixed" subgroup H):

gap> G := SymmetricGroup(4);;
gap> ccsg := ConjugacyClassesSubgroups(G);;
gap> H := Representative(ccsg[5]);;   # the Klein-4 group
gap> intHG := IntermediateSubgroups(G,H);
rec( subgroups := [ Group([ (1,4)(2,3), (1,3)(2,4), (3,4) ]), Group([
(1,4)(2,3), (1,3)(2,4), (2,3) ]),
      Group([ (1,4)(2,3), (1,3)(2,4), (2,4) ]), Group([ (1,4)(2,3),
(1,3)(2,4), (2,4,3) ]) ],
  inclusions := [ [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 0, 4 ], [ 1, 5 ], [
2, 5 ], [ 3, 5 ], [ 4, 5 ] ] )

The subgroups field of the intHG structure gives the list of subgroups
of G that contain H.  You can access the i-th intermediate subgroup
using intHG.subgroups[i].   (Note that H and G are not included in the
intHG.subgroups list.)  The inclusions field gives the lattice of
intermediate subgroups by showing the covering relations.

-William





On Sun, May 13, 2012 at 9:44 PM, Mr. Sorouhesh <msorouhesh at gmail.com> wrote:
> Suppose we have a finite group such that know all its subgroups. Now,
>   fix a certain subgroup in the group.Can we use GAP to list all
>   subgroups of the group that contain our fixed subgroup?
>   Best
>
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-- 
William J. DeMeo
Department of Mathematics
University of Hawaii at Manoa
phone: 808-298-4874
url: http://math.hawaii.edu/~williamdemeo