[GAP Forum] About Normalizer

[Joe Bohanon]

In this case, PSL(2,9) is small enough to just construct the entire lattice,
so you could do:

gap> Filtered(List(ConjugacyClassesSubgroups(G),Representative),i->Size(i)=8
and IsDihedralGroup(i));
[ Group([ (3,4)(5,8)(6,9)(7,10), (3,6,4,9)(5,7,8,10), (1,2)(5,10)(6,9)(7,8)
]) ]
gap> Normalizer(G,last[1]);
Group([ (3,4)(5,8)(6,9)(7,10), (3,6,4,9)(5,7,8,10), (1,2)(5,10)(6,9)(7,8) ])

Now, let's say you have a bigger group for which this is not possible. Given
your setup, you can run:

List(IsomorphicSubgroups(G,H),Image);

to get all of the subgroups you want. Isomorphic subgroups returns
isomorphisms, so you could also start with a representation of H that you
like, and run it on that.

For several large simple groups there's also a table of marks library that
has things like subgroup lattices and maximal subgroups already stored.

For a group where you can still get conjugacy classes, you can also just
take random pairs of involutions (actually, you'd just fix one from a
conjugacy class and randomize the other one), until their product has order
4.

There are ways to still get this to work in groups too big for
ConjugacyClasses to successfully run given your memory assuming you can
still find (pseudo)-random elements.

Of course, you can tweak all this to your needs for other subgroups.

Joe

On Wed, May 11, 2011 at 10:14 AM, Dung Duong <dungdailoan at gmail.com> wrote:

> Dear GAP Forum,
>
> I am totally new with GAP and I have a problem as below :
>
> I have a group G:=AutomorphismGroup(PSL(2,9)). One of its Maximal subgroup
> is $M =N_G(D_8)$. And I would like to computthe order of this group. So I
> did as follows :
>
> > G:=AutomorphismGroup(PSL(2,9));
> > H:=DihedralGroup(8);
> > Normalizer(G,H);
>
> But it does not work. I think that somehow I need to "embed" H as a
> subgroup
> of G. But I dont know how.
>
> Please tell me how I can do.
>
> Thanks in advance.
>
> Best regards,
>
> Dung Duong
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