> < ^ Date: Fri, 25 Jan 2002 12:11:32 +0100 (CET)
> < ^ From: Joachim Neubueser <joachim.neubueser@math.rwth-aachen.de >
> ^ Subject: Re: Subgroup question (fwd)

Daer Gap Forum members,

In a Forum mail answering a question of Igor Schein, I had written:

There is no special function in GAP which would directly answer the
question if a given group contains a subgroup isomorphic to the
quaternion group, or that would just find such a subgroup.

I want to thank Stefan Kohl for correcting me in a private letter with
respect to this statement. As he pointed out, in chapter 37 of the
reference manual the following operation is described::

> IsomorphicSubgroups( <G>, <H> )  

computes all monomorphisms from <H> into <G> up to <G>-conjugacy of
the image groups. This classifies all <G>-classes of subgroups of <G>
which are isomorphic to <H>.

With the existing methods the amount of time needed grows with the
size of a generating system of <G>. (Thus in particular for $p$-groups
calculations can be slow.)

If the `findall' option is set to `false', the algorithm will stop
once one homomorphism has been found (this can be faster and might be
sufficient if not all homomorphisms are needed).

gap> g:=Group((1,2,3,4),(1,2));
Group([ (1,2,3,4), (1,2) ])
gap> h:=Group((3,4),(1,2));;
gap> emb:=IsomorphicSubgroups(g,h);
[ [ (3,4), (1,2) ] -> [ (3,4), (1,2) ],
  [ (3,4), (1,2) ] -> [ (1,3)(2,4), (1,2)(3,4) ] ]

While likely, if one wants to look for quaternion groups in a large
number of groups, a small program along the suggestions in my last
letter will be faster, this operation certainly can be used.

Thanks again to Stefan for carefully reading the Forum messages.

Joachim Neubueser

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