Purely definable subgroup

Definition
A subgroup of a group is said to be purely definable if it is definable as a subset in the first-order theory of the pure group. By pure group, we mean the set equipped only with the structure of the group operations and with no additional first-order data supplied.

By definable subset, we mean that there is a first-order formula with one free variable such that the set of elements satisfying the formula is precisely that subset.