> < ^ From:

> ^ Subject:

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

> < [top]