> < ^ Date: Tue, 12 Nov 1996 12:47:21 +0100
> < ^ From: Bettina Eick <beick@tu-bs.de >
> < ^ Subject: Re: aut

Dear gap forum,

Pat Callahan wrote:
> How do you get GAP to compute the automorphism group of a group?

In general, you could use the function `AutomorphismGroup(G)' to compute
the automorphism group of the group G.

If your group happens to be a finite p-group given as AgGroup, then you
could also use the function `AutomorphismsPGroup' contained in the share
package ANUPQ. This will be much more efficient than the generic function.
It will return a list of automorphisms which generate a supplement to the
inner automorphism group in the full automorphism group. To call this
function you have to `RequirePackage("anupq")' first.

(In near future we expect that there will also be another share package
released which will contain efficient functions to compute the automorphism
group of a finite soluble group.)

> I am interested in the question: What is the smallest group which
> has an automorphism group of odd order?
The trivial group, of course :-)
Next there is the cyclic group of order 2.

The smallest non-trivial group with automorphism group of odd order
has size 3^6. There is a catalogue of all 3-groups of order up to
3^6 in GAP. You can get a group of this catalogue by the command
ThreeGroup( size, nr ). The group ThreeGroup( 3^6, 31 ) has an
automorphism group of odd order.

I checked that none of the groups of order between 3 and 3^6-1 has
an automorphism group of odd order using a catalogue of all groups
of order up to 1000 without 512 and 768 and using GAP to compute
automorphism groups. This catalogue has been computed just recently
by Hans Ulrich Besche and myself and it is not publically available
yet.

To find the smallest group with automorphism group of odd order
it is enough to consider groups of odd order.
(Consider the Sylow 2-subgroup S of a group G. Either S is not
contained in the centre of G, then 2 divides the size of the inner
automorphism group. Or S is contained in the center, then it is
normal and thus has a complement K in G. Therefore G = S x K and
Aut(G) = Aut(S) x Aut(K). Hence Aut(K) has odd order as well and
S must be trivial or the cyclic group of order 2.)

Best wishes,

Bettina

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