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

Suppose $G$ is a nontrivial group that is a 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 Torsion-free group (?) (i.e., all the elements of $G$ have infinite order) or $G$ is a Group of prime exponent (?) (i.e., there exists a prime number $p$ such that every non-identity element of $G$ has order $p$).