Automorphism group is transitive on non-identity elements implies aperiodic or prime exponent

Statement
Suppose $$G$$ is a nontrivial group that is a fact about::group whose automorphism group is transitive on non-identity elements. In other words, for any two non-identity elements $$g,h \in G$$, there is an automorphism $$\sigma$$ of $$G$$ (i.e., an element of the automorphism group $$\operatorname{Aut}(G)$$) such that $$\sigma(g) = h$$.

Then, either $$G$$ is a fact about::torsion-free group (i.e., all the elements of $$G$$ have infinite order) or $$G$$ is a fact about::group of prime exponent (i.e., there exists a prime number $$p$$ such that every non-identity element of $$G$$ has order $$p$$).

(For an abelian group, the condition that the automorphism group is transitive on non-identity elements is equivalent to the condition that the group is the additive group of a field. In particular, it is either an elementary abelian group or a vector space over the field of rational numbers. See additive group of a field for more.)

Related facts

 * Baer's theorem on elation groups