Finite and automorphism group is transitive on non-identity elements implies elementary Abelian

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Statement

Suppose G is a finite group whose automorphism group acts transitively on the non-identity elements of G. In other words, G is a Group whose automorphism group is transitive on non-identity elements (?). Then, G is either trivial or an elementary Abelian group: it is a direct product of cyclic groups of order p for some prime p.

Related facts

Breakdown for infinite groups

There exist infinite groups whose automorphism group is transitive on the non-identity elements, that are not even Abelian.

Other related facts