# Group whose automorphism group is transitive on non-identity elements

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## Definition

### 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.