I wanted to ask what is known about finite groups whose automorphism group is decomposable (i.e. $Aut(G)=H \times K$ for some groups $H,K$) ? Is there for instance some kind of classification or do such groups share some interesting properties ?

The only general fact I know about the subject is the well-known fact that if $H,K$ are finite groups of coprime order then $Aut(H \times K) =Aut(H) \times Aut(K)$.

  • 1
    $\begingroup$ Consider just the case of cyclic groups. The cyclic groups with this property are those whose order is neither 2 nor a Fermat prime. $\endgroup$
    – Will Sawin
    Apr 21, 2012 at 8:24
  • 2
    $\begingroup$ More generally, if $H$ and $K$ are groups for which ${\rm Hom}(H,K)$ and ${\rm Hom}(K,H)$ are both trivial then ${\rm Aut}(H \times K) = {\rm Aut}(H) \times {\rm Aut}(K)$. $\endgroup$
    – Derek Holt
    Apr 21, 2012 at 13:20
  • $\begingroup$ @Will and Derek: These are interesting examples I wasn't aware of so far. Many thanks. $\endgroup$ Apr 21, 2012 at 18:37
  • $\begingroup$ Also, if $T$ is a finite nonabelian simple group then $Aut(T^n)\cong Aut(T)\wr S_n$. More generally, the structure of $Aut(H_1\times \cdots \times H_n)$ for arbitrary finite groups $H_i$ is described by Jonni Bidwell in his Ph.D. thesis. Though it won't answer your question, you can find this work at: springerlink.com/content/ch7370l4165l419r/?MUD=MP. $\endgroup$ Apr 25, 2012 at 10:37


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.