# on a group with perfect automorphism group

A group $G$ is called perfect if $G=G'$.

Does there exist a group $G$ such that $Aut(G)$, the automorphism group of $G$, is perfect?

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What made you to think such these kinds of non trivial groups may exist? – Babak S. Apr 19 '13 at 8:17
Hint: Do you know what outer automorphisms are? If yes, look for a simple group with trivial outer automorphisms group. – j.p. Apr 19 '13 at 8:46

Let $G=\mathbb{Z}_2^n$ with $n>2$. Then $Aut(G)=GL(n,2)=SL(n,2)$ is perfect.
I know the next definition: a group $G$ is perfect if $G$ has a trivial center and each automorphism of $G$ is inner. This suggests that $Aut(G)$ is isomorphic to $G$ for each perfect group $G$. In particular, for the group $S(X)$ of all bijections of a set $X$, where $|X|\ge 3$ and $|X|\not=6$. Moreover, W. Specht obtained the following result. Let $G$ be a group $G$ without center and $In(G)\le Aut(G)$ be a group of all inner automorphisms of the group $G$. If $\alpha(In(G))=In(G)$ for each automorphism $\alpha$ of the group $Aut(G)$, then $Aut(G)$ is perfect.