Welcome to MathOverflow

Question

5

3

There are two non-abelian groups of order p^3, namely, semi-direct product of Z/pZ x Z/pZ by Z/pZ and semi-direct product of Z/(p^2)Z by Z/pZ. What are the automorphism groups of these groups?

flag	asked Jan 21 2011 at 5:01 RDK 421●1●8

1

Have you tried asking GAP? – Mariano Suárez-Alvarez Jan 21 2011 at 5:11

In GAP, I tried for first group with p=3,( i.e. G=Z/3Z x Z/3Z): Z/3Z); but no output. It gives |Aut(G)|=432, but no structure description. So I am looking theoretically. I did following: since any automorphism of G fixes center(not necessarily pointwise), there is natural homomorphism f:Aut(G) --> Aut(G/Z(G)), where |Z(G)|=p, and G/Z(G)=Z/pZ x Z/pZ, Aut(G/Z(G))=GL(2,p). We can show that f is surjective with kernel Z/pZ x Z/pZ. Hence |Aut(G)|=(p^2).|GL(2,p)|=(p^3).(p+1).(p-1)^2. I couldn't proceed further. – RDK Jan 21 2011 at 6:02

1

It's instructive to work out such examples from scratch, but it's also important to realize that this has certainly been done before and is most likely written down in detail (somewhere). With over a century of literature on group theory including many books, it could take a while to locate such information; maybe it's quicker to do it yourself. Anyway, it's also useful to organize the possible methods such as generators and relations, linear representations, geometric realizations. GAP has its limitations for giving insight. – Jim Humphreys Feb 22 2011 at 15:32

This example I couldn't see in any Graduate Studies book; it is not also given as an exercise. Late, I found that it is an exercise in the book "Structure of Groups of Prime Power Order". When working such examples in GAP, it gives the order of Automorphism group of these groups quickly, but it is not giving information about structure. – RDK Feb 23 2011 at 3:55

@Rahul: I deleted my answer (I think), and instead typed everything up into a PDF. You can email me and I'd be glad to send it to you. – Steve D Feb 23 2011 at 9:11

Chris Gerig · Accepted Answer · 2011-01-21 05:42:02Z UTC

7

For the latter group, the answer is Bidwell & Curran's paper "The Automorphism Group of a Split Metacyclic p-Group"

http://www.springerlink.com/content/c7324108h86282l1/

answered Jan 21 2011 at 5:42

Chris Gerig
5,536●1●12●40

2

see also: groupprops.subwiki.org/wiki/… – Igor Rivin Jan 21 2011 at 15:27

Tom De Medts · Answer 2 · 2011-02-22 08:29:08Z UTC

The former group can be seen as the group of unitriangular $3 \times 3$-matrices over the field with $p$ elements: $$G = \left\{ \begin{pmatrix} 1 & * & * \\ 0 & 1 & * \\ 0 & 0 & 1 \end{pmatrix} \right\} \leq SL(3,p)$$ The automorphism groups of such groups have been studied (in a much larger generality); see for instance the paper "The automorphism group of the group of unitriangular matrices over a field" by Ayan Mahalanobis (http://arxiv.org/abs/1012.5534v1).

Thomas · Answer 3 · 2012-11-07 01:03:39Z UTC

3

The automorphism groups of all p-groups of order p^3 can be found at http://www.math.kth.se/~boij/kandexjobbVT11/Material/pgroups.pdf

answered Nov 7 at 1:03

Thomas
31●1

Automorphisms of non-abelian groups of order p^3

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

3 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.