In general, the center of an algebraic structure is the set of elements which commute with every of the structure. With magmas (such as groups), this definition is straightforward; for rings and fields, the commutativity in question is multiplicative commutativity.
The center of a group is never empty, as the identity commutes with every element of a group. The center of a group is a subgroup of the group—a normal subgroup, in fact; it is also stable under any endomorphism on the group.
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