Center is purely definable

Statement
The center of a group is a purely definable subgroup: it can be defined as a subset in the first-order theory of the pure group.

Proof
We provide here the formula $$\varphi$$ that an element $$x \in G$$ satisfies if and only if it is in the center:

$$\varphi(x) = \ \forall \ y \in G : xy = yx$$

This is a first-order description, so the center is purely definable.