Central implies abelian normal

Statement
Any central subgroup of a group (i.e., any subgroup contained in the center of the group) is an abelian normal subgroup (i.e., it is both a normal subgroup and an abelian group).

Related facts

 * Abelian normal not implies central
 * Central factor implies normal
 * Normal not implies central factor

Facts used

 * 1) uses::Central implies normal
 * 2) uses::Central implies abelian

Proof
The proof follows from facts (1) and (2).