Definable subgroup

Definition
A subgroup of a group (possibly with additional structures and relations) is said to be definable if the membership test for the subgroup can be expressed in first-order language in terms of the theory of the group.

Facts
In the case of a pure group (viz a group without any additional structure), the definable subgroups are those arising from first-order subgroup-defining functions. Incidentally, the definable subgroups in that case are also always elementarily characteristic subgroups. However, when we introduce additional structure, the definable subgroups may not be elementarily characteristic, or even characteristic.