Center is elementarily characteristic

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., elementarily characteristic subgroup)}
View subgroup property satisfactions for subgroup-defining functions $|$ View subgroup property dissatisfactions for subgroup-defining functions

Statement

The center of a group is an elementarily characteristic subgroup: there is no other subgroup of the group that is elementarily equivalently embedded.