Show that if and are both cyclic then .

We begin by proving a lemma.

Lemma: Let be a group and let be normal. Then . Proof: If is a commutator, then . , so that is abelian. By Theorem 5.7 in the text, .

Now for the main result.

Note that is characteristic, hence normal, so that is a cyclic, hence abelian, normal subgroup.

By this previous exercise, (via the Third Isomorphism Theorem) acts on by conjugation on the left as follows: . Now conjugation is an automorphism of , and since this quotient is cyclic, its automorphism group is abelian. Thus, for all and , we have . Via some arithmetic, we see that ; in particular, each element of commutes with each generator of , so that .

Using the Third Isomorphism Theorem, is cyclic; hence is abelian, and using the lemma we have . Thus .