Take the 2-minute tour ×  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?  share|improve this
question Have you tried asking GAP? –  Mariano Suárez-Alvarez
Jan 21 '11 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 '11 at 6:02 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