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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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