[GAP Forum] About certain mixed products of cyclic groups

[Derek Holt]

Dear GAP Forum, Hebert,

On Tue, May 11, 2010 at 11:23:15AM +1000, Hebert Pérez-Rosés wrote:
> Dear all,
> 
> I want to construct all the groups of type (Zm x Zm) \rtimes Zn for given
> m,n, but I don't know how to do that in GAP. In relation to that, there
> are a few sub-problems (some of a rather group-theoretical nature) for which
> I don't know the answer. I will appreciate anyone pointing me to the right
> references:
> 
> 1. What is the group of automorphisms of Zm x Zm, for arbitrary m?

Bettina suggested that a theoretical analysis of this automorphism group
might help with computing its conjugacy classes.

Aut(Zm x Zm) is the group GL(2,Z_m) where Z_m is the ring of integers mod m.
It is the direct product of Aut(P) where P ranges over the Sylow subgroups
of Zm, so you should be able to reduce to the case when m is a prime power.
When m = p^k for prime k, Aut(Zm x Zm) is an extension of a normal p-subgroup
of order p^{4(k-1)} by GL(2,p).

You can create this group in GAP as follows:

gap> R:=ZmodnZ(27);
(Integers mod 27)
gap> G:=GL(2,R);
GL(2,Z/27Z)
gap> Order(G);
314928
gap> C:=ConjugacyClasses(G);;        
gap> Length(C);
720

Unfortunately, this seems to be no quicker that computing Aut(Z27 x Z27)
directly and then computing its conjugacy classes, but I would expect the
reduction to m being a prime power should definitely help to speed things up.

Derek Holt.

> 2. How can I express an arbitrary automorphism of Zm x Zm by its action on
> the abstract generators "a" and "b" of Zm x Zm ?

> 3. How can I get a pc presentation of (Zm x Zm) \rtimes Zn in GAP?
> 
> Hebert Perez-Roses
> The University of Newcastle, Australia.
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