# Automorphism group is transitive on non-identity elements implies aperiodic or prime exponent

The statement of this article contains an assertion of the form that for a certain kind of group, either every element has finite order (i.e., the group is a Periodic group (?)) or every non-identity element has infinite order (i.e., the group is a Torsion-free group (?) or aperiodic group). The actual statement in this case may be considerably stronger.

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## Statement

Suppose is a nontrivial group that is a Group whose automorphism group is transitive on non-identity elements (?). In other words, for any two non-identity elements , there is an automorphism of (i.e., an element of the automorphism group ) such that .

Then, either is a Torsion-free group (?) (i.e., all the elements of have infinite order) or is a Group of prime exponent (?) (i.e., there exists a prime number such that every non-identity element of has order ).

(For an abelian group, the condition that the automorphism group is transitive on non-identity elements is equivalent to the condition that the group is the additive group of a field. In particular, it is either an elementary abelian group or a vector space over the field of rational numbers. See additive group of a field for more.)