The title basically says it all.


Is there a group with more than one element that is isomorphic to the group of automorphisms of itself?


I'm mainly interested in the case for finite groups,
although the answer for infinite groups would still be somewhat interesting.



Is there a non-trivial group G isomorphic to Aut(G)?
The automorphism group of the symmetric group $S\_n$ is (isomorphic to) $S\_n$ when $n$ is different from $2$ or $6$. In fact, if $G$ is a **complete** group you can ascertain that $G \simeq \mathrm{Aut}(G)$. The reverse implication needn't hold, though.


