Purely omega-categorical group

Definition
A group is said to be purely omega-categorical if it is countable and is omega-categorical as a pure group. In other words, a group is said to be omega-categorical if it is countable and if its automorphism group has only finitely many oribts in any direct power of the group.

Stronger properties

 * Finite group

Weaker properties

 * Possibly omega-categorical group
 * Nilpotent-by-finite group

Facts
Every characteristic subgroup of a purely omega-categorical group is purely definable.