Central implies abelian normal

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., central subgroup) must also satisfy the second subgroup property (i.e., abelian normal subgroup)
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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. Central implies normal
2. Central implies abelian

Proof

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