Purely omega-categorical group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This article defines a group property obtained by applying the purely operator (viz taking the pure group version) of the following property of groups with additional structure: 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.

Relation with other properties

Stronger properties

Weaker properties

Facts

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