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Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties

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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 , there exists such that σ(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