Groups with transitive automorphisms Let $G$ be a finite group such that for each $a,b \in G \setminus \{e\}$ there is an automorphism $\phi:G \rightarrow G$ with $\phi(a)=b$. Prove that $G$ is isomorphic to $\Bbb Z_p^n$ for some prime $p$ and natural number $n$.

Hint 1: If $a, b \in G \setminus \{e\}$, then $a$ and $b$ have the same order.
Hint 2: Using the previous hint, show that $G$ has order $p^n$ for some prime $p$ and that every nonidentity element has order $p$. 
Hint 3: In a $p$-group, the center is a nontrivial characteristic subgroup.
