Group whose automorphism group is transitive on non-identity elements

Symbol-free definition
A group whose automorphism group is transitive on non-identity elements is a group with the property that given any two non-identity elements of the group, there exists an automorphism of the group sending the first to the second.

Definition with symbols
Let $$G$$ be a group. Then we say that the automorphism group of $$G$$ is transitive on non-identity elements if, given any two non-identity elements $$g,h \in G$$, there exists $$\sigma \in \operatorname{Aut}(G)$$ such that $$\sigma(g) = h$$.

Note that for an Abelian group, this is equivalent to the property of being the additive group of a field.

Stronger properties

 * Weaker than::Additive group of a field
 * Weaker than::Group with two conjugacy classes

Weaker properties

 * Stronger than::Group in which every element is order-automorphic
 * Stronger than::Group in which any two elements generating the same cyclic subgroup are automorphic
 * Stronger than::Group in which every element is automorphic to its inverse
 * Stronger than::Characteristically simple group