Omega-categorical group

Symbol-free definition
A group (possibly with additional structures and relations) is said to be omega-categorical if it is countable and satisfies the following equivalent groups:


 * For every natural number $$n$$, the group of automorphisms of the group (which preserve all the additional structure) have only finitely many orbits in its $$n^{th}$$ power.
 * For any $$n$$-tuple of elements, the action of the stabilizer of that tuple has only finitely many orbits in the group.

The property of being omega-categorical can be evaluated, not just for groups with additional structure, but for any first-order structure.

Stronger properties

 * Stable omega-categorical group

All characteristic subgroups are definable
In an omega-categorical structure, a set is definable with parameters when it is preserved by all automorphisms of the group. Thus, in particular, any characteristic subgroup of the group is definable. (Characteristic subgroups, being preserved by all group automorphisms, are in particular preserved by those automorphisms that also preserve the additional structure).