> < ^ Date: Tue, 09 Mar 1993 04:51:58 +0100
> ^ From: Mark Short <short@jordan.murdoch.edu.au >
^ Subject: Recognising small groups

Ralf Dentzer writes (in part):
> The groups I am interested in are not very big, at most some hundred
> elements, but can be rather complicated in structure.
> My first problem is to recognize some groups from the literature or other
> computations in the complete lists of (small) solvable groups (or
> p-groups) contained in GAP, so that I get a complete overview of
> the possibilities. Some of the groups are given by relations, some
> as permutation groups and some as semidirect products or otherwise.
>
> The second problem is to recognize subgoups and/or factor groups of
> (solvable) groups in the given lists or to test isomorphism with
> other subgroups/factor groups. Especially the isomorphism test
> between subgroups should happen automatically (the groups involved
> in this are very small!)

The good news: I have code that recognises many ``small'' groups. For example,
if given GL(2,3) in any form, then (assuming any coset enumerations that might
be needed can be accomplished, and so on) it would return something like this:

```Order: 48
Identifier: 48.49
Name: GL(2,3)
```

I won't bother to describe any more details because of the following items of

1. The code is written in Cayley. GAP didn't have all the features I needed
when I first started the project in 1987. I think GAP has all the necessary
features now, but the code comprises 3000 lines, so I am in no hurry to make
the conversion!
So, if you can be bothered you might want to make part of this conversion
yourself.

2. My identifiers for the groups are usually not those used by GAP. For example,
the group A_4 would come back as

```Order: 12
Identifier: p^2q # 2g
Name: A(4)
```

(This is because an analogue of A_4 exists for many orders p^2q (where p and q
are distinct primes) and my code can recognise any such group.)
If your group has a name, then this is no problem, but if it doesn't, then you
just get an identifier, which of course is no help in recognising the group
unless you know my classification scheme!

```-----
```

So Ralf, yes I can help you, but only if you are willing to put in a
significant amount of work!

If you want any more details please write to me directly at
short@jordan.csu.murdoch.edu.au