# [GAP Forum] Subgroups of a given order

Laurent Bartholdi laurent.bartholdi at gmail.com
Fri Jul 11 12:26:58 BST 2008

```Hi Josef,
If your group is small enough, you can list all subgroups by
constructing all conjugacy classes. This is usually impractical for
large groups, so you need another method. You may for instance compute
orders of elements to see if there is a cyclic subgroup of desired
size; or start by a cyclic subgroup, compute its normalizer, and try
adding elements of the normalizer to create by successive extensions a
subgroup of desired size.

For the first, most naive appoach, here's an example session:
gap> ConjugacyClassesSubgroups(SymmetricGroup(5));
[ Group( () )^G, Group( [ (4,5) ] )^G, Group( [ (2,3)(4,5) ] )^G,
Group( [ (3,4,5) ] )^G,
Group( [ (2,3)(4,5), (2,4)(3,5) ] )^G, Group( [ (2,3)(4,5), (2,4,3,5) ] )^G,
Group( [ (4,5), (2,3) ] )^G, Group( [ (1,2,3,4,5) ] )^G, Group( [
(3,4,5), (4,5) ] )^G,
# < leaving out 4 lines ...  >
, Group( [ (1,2,3,4,5), (3,4,5) ] )^G, SymmetricGroup( [ 1 .. 5 ] )^G ]
gap> Concatenation(List(last,Elements));
[ Group(()), Group([ (4,5) ]), Group([ (3,4) ]), Group([ (3,5) ]),
Group([ (2,3) ]), Group([ (2,4) ]),
Group([ (2,5) ]), Group([ (1,2) ]), Group([ (1,3) ]), Group([ (1,4)
]), Group([ (1,5) ]),
Group([ (2,3)(4,5) ]), Group([ (2,4)(3,5) ]), Group([ (2,5)(3,4) ]),
Group([ (1,2)(4,5) ]),
# < leaving out 49 lines ...  >
Group([ (1,2,3,4,5), (3,4,5) ]), Group([ (1,2,3,4,5), (1,2) ]) ]
gap> Filtered(last,x->Size(x)=6);
[ Group([ (3,4,5), (4,5) ]), Group([ (2,4,5), (2,5) ]), Group([
(1,4,5), (1,5) ]),
Group([ (2,3,4), (3,4) ]), Group([ (1,3,4), (3,4) ]), Group([
(2,3,5), (2,3) ]),
Group([ (1,3,5), (1,3) ]), Group([ (1,2,3), (2,3) ]), Group([
(1,2,4), (2,4) ]),
# < leaving out 6 lines ...  >
Group([ (2,4), (1,3,5) ]), Group([ (1,3), (2,4,5) ]), Group([ (2,5),
(1,3,4) ]) ]
gap> Filtered(last,x->Size(x)=7);
[  ]

On Fri, Jul 11, 2008 at 11:53 AM, Josef Lauri <josef.lauri at um.edu.mt> wrote:
> I know this must be a silly question, but here goes: Given a group G how can
> I out find if G has subgroups of given order k and how can I list them if
> there are any?
>
> Thanks.
>
> Josef Lauri
>
> _______________________________________________
> Forum mailing list
> Forum at mail.gap-system.org
> http://mail.gap-system.org/mailman/listinfo/forum
>

--
Laurent Bartholdi \ laurent.bartholdi<at>gmail<dot>com
EPFL SB SMA IMB MAD \ Téléphone: +41 21-6935458
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Home address: http://microurl.org/10, http://microurl.org/16

```