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

From Groupprops

## Statement

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

## 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.

- Equivalence of definitions of additive group of a field: For an Abelian group, the automorphism group being transitive on non-identity elements is equivalent to the group being the additive group of some field.