The problem is to
Show that if $G$ is a finite group and for all nontrivial elements $a, b$ there exists an automorphism taking $a$ to $b$, then $G$ is a $C_p$ vector space, where $C_p$ is the group of prime order $p$.
My question is if an additional hypothesis that $G$ is abelian is needed.
I cannot seem to prove that $G$ is abelian from the hypotheses.
Of course, the point is that all the elements of $G$ have the same prime order $p$.
But I cannot seem to get the result without showing $G$ is abelian, in which case the normality of all subgroups gives what I want.
We may assume $G$ is not trivial. As mentioned in the statement of the problem, it is quick to go from the hypotheses to the fact that $G$ is a $p$-group. Now, every nontrivial $p$-group has a nontrivial center. Let $z$ be some nonzero element of the center. Given $g ,h \in G$, by hypothesis we may choose an automorphism $\phi$ carrying $g$ to $z$. Then, $\phi(gh) = \phi(g)\phi(h) = \phi(h)\phi(g) = \phi(hg)$. Since $\phi$ is injective, we see that $gh = hg$. So, $G$ is abelian.