Center is purely definable

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) always satisfies a particular subgroup property (i.e., purely definable subgroup)}
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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.

Related facts

Weaker facts: weaker subgroup properties satisfied

Property	Meaning	Proof that it is satisfied by the center
elementarily characteristic subgroup	no other elementarily equivalently embedded subgroups	center is elementarily characteristic
characteristic subgroup	invariant under all automorphisms	center is characteristic
normal subgroup	invariant under all inner automorphisms	center is normal

Similar subgroup-defining functions being purely definable

Subgroup-defining function	Meaning	Relation to center	Proof that it is purely definable
member of the finite upper central series	ascending series obtained starting from trivial group where each member's quotient by predecessor equals center of group's quotient by predecessor	center is first nontrivial member	finite upper central series member is purely definable

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.