TITLE: Homogeneous groups
QUESTION [9 upvotes]: Let's call a group $G$ homogeneous if for every two distinct, non-identity elements $a$ and $b$ there is an automorphism $\phi$ of $G$ such that $\phi(a)=b$.
Examining this definition, we can see that the underlying additive group of any field (indeed, any division ring) is homogeneous by examining the automorphism $\phi(x)=ba^{-1}x$.
Thus for any $n$ and any prime $p$, we have that $Z_p^n$ is homogeneous. Conversely, if we also know that $G$ is finite, we can show that $G$ is of the form $Z_p^n$ for some $n$ and $p$.

We can clearly see that all elements of a homogeneous group have the same order. And by Cauchy's theorem, we get that $G$ must have prime-power order. Say $|G|=p^n$. The Conjugacy Class Equation then implies that $G$ has non-trivial center. And since $G$ is homogeneous, all elements are in the center and thus $G$ is abelian. Since $G$ is finite, abelian, and all elements have the same order, we can conclude that $G$ is isomorphic to $Z_p^n$. If it weren't the fundamental theorem of abelian groups would imply there were an element of order $p^i$ for $i>1$ contradicting the fact that all elements have the same order (Cauchy's theorem still implies there's an element of order $p$).

Thus finite, homogeneous groups are kind of boring. Infinite examples are more interesting as we can garner a couple examples just from fields, and they are more difficult to investigate. We can still surmise a couple things when $G$ is infinite. For example, if $G$ has a single non-identity element of finite order then all (non-identity) elements of $G$ have prime order.

Let $x\neq e$ have finite order $n$. Let $n=p^em$ for some prime $p$ with $e\geq 1$ and $p\nmid m$. Then the element $x^{p^{e-1}m}$ has order $p$, and thus all elements have order $p$.

Also, in general, 

If $G$ has a non-trivial center then it is abelian.
Any quotient by a characteristic subgroup is also homogeneous.

I am having difficulty figuring out other, even basic, things though. My question is kind of general:

What else can we determine about homogeneous groups?

I am specifically interested in these questions:

Does $G$ have to be abelian?
If not, can $G$ be perfect?
Can $G$ be free? It clearly can't have rank 1, but what about other ranks?
If all elements have prime order, is $G$ a direct sum of multiple copies of $Z_p$?

But any addition is welcome and appreciated.
-UPDATE
Qiaochu Yuan has deftly answered three of my original questions, but one still remains though it may be difficult. However his answer has left me intrigued for a proof of the following claim:

Every abelian, homogenous group is the underlying additive group of some field.

I don't see how you could construct the multiplication operation, but it seems the choice for which element would become unity is a free choice.

REPLY [2 votes]: You may be interested in this paper. The author proves that, in a certain sense, free groups of finite rank are homogeneous. The "certain sense" is, loosely, that if $a$ and $b$ satisfy the same formulae then they are connected by an automorphism.