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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
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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

2 Answers 2

up vote 8 down vote accepted

Let $G=\mathbb{Z}_2^n$ with $n>2$. Then $Aut(G)=GL(n,2)=SL(n,2)$ is perfect.

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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.

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What you define here as a "perfect group" is called a complete group (wikipedia). A group is called perfect when it equals its commutator subgroup (mathworld). These are two different things. –  Mikko Korhonen Apr 19 '13 at 10:03
    
@ Alex Ravsky; Please write full title of Specht paper. –  maryam Apr 19 '13 at 10:08
    
W. Specht. Gruppentheorie, Berlin-Göttingen-Heidelberg, 1956. –  Alex Ravsky Apr 19 '13 at 10:16

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