# Purely omega-categorical group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

*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

## Contents

## 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.