How Aut(G) acting transitively on a finite group G^* can lead G to be elementary abelian group? G^*=G{1}.
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A way to see this is: a. An automorphism maps the center of $G$ to the center of $G$. Thus, under your condition the center of $G$ is the full group and it is thus abelian, or the center is trivial. b. Note that an automorphism preserves the order of an element. Thus the transitivity implies that all elements of $G$ except $1$ have the same order $n$. And this $n$ (thus) has to be prime. In particular, the center is nontrivial. Thus, $G$ is a finite abelian $p$group in which all elements (except $1$) have order $n$ and $n$ is prime. That is, it is an elementary abelian pgroup. Note: The other answer appeared while this was basically typed; I still post it as perhaps the additional details are useful. 


G must have exponent p where p is prime; and G must be characteristically simple. These are both easy arguments. Isn't that enough? 

