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.

Relation with other properties

Stronger properties

• Additive group of a field
• Group with two conjugacy classes

Weaker properties

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