From: Derek Holt 
Date: Fri, 17 Apr 1998 04:05:51 +0100
Subject: Automorphism of a Frattini extension


Hallo pubbers!

Can anyone help me with the following technical question about group
extensions?

Let  1 -> M -> G -> Q -> 1  be a group extension, with M elementary
abelian and M contained in Frattini(G), G finite.

Does an automorphism of G fixing M that induces the indentity on Q
necessarily restrict to the identity on M?

I have been assuming that the answer is yes, but I have suddenly had
doubts!

There is an old result of Gaschuetz that, for a given Q and prime p
there is a unique maximal extension of this kind for which M is an
elementary abelian p-group and M is contained in the Frattini subgroup
of G (which means that the extension is completely non-split), of
which all other such extensions are quotients.

Derek Holt.

------ and then soon after---

Date: Fri, 17 Apr 1998 00:54:06 -0400 (EDT)
From: mpettet@uoft02.utoledo.edu
Subject: Re: Automorphism of a Frattini extension


The answer is certainly "yes" if the automorphism is of order coprime to
the order of G but, unless I have misunderstood the question, it can't 
be true in general. Take G to be a finite p-group whose Frattini subgroup
M is elementary abelian but not central and take the automorphism to be
the inner automorphism induced by any element which doesn't centralize the
Frattini subgroup. If p is at least 3, I think Z_p wreath Z_p will do for 
G. Am I missing something?

Martin Pettet

----- and then Derek clarified his question-----

Date: Fri, 17 Apr 1998 06:24:52 +0100

Sorry I must have been in the pub too long. Martin Pettet and others have
quickly pointed out that the answer is certainly no.

I would like to make the additional assumption that G acts irreducibly on the
module M, or better, that M is a direct sum of irreducible G-modules.

Derek Holt.