Theorem: Let be a group and let be the center of . If is cyclic, then is Abelian.
Proof: Suppose that is cyclic. Then there is some such that .
Let . We wish to show that . It follows that there exist integers such that and . So there exist such that and . Now consider their product:
Note that in the third line above we are able to commute and since . Similarly we are able to commute from line four to five.
Thus, we have shown that is Abelian.
Reflection: It is necessary that your subgroup is the center of , otherwise we wouldn’t be able to commute and conclude that is Abelian. The proof itself followed mainly from definitions and helpful manipulation.