Group whose automorphism group is transitive on non-identity elements
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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 be a group. Then we say that the automorphism group of is transitive on non-identity elements if, given any two non-identity elements , there exists such that .
Note that for an abelian group, this is equivalent to the property of being the additive group of a field.