Center is purely definable

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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)}
View subgroup property satisfactions for subgroup-defining functions View subgroup property dissatisfactions for subgroup-defining functions

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 that an element satisfies if and only if it is in the center:

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