Let be a group and .
- Show that . Give an example to show that this is not necessarily true of is not a subgroup.
- Show that if and only if is abelian.
First we prove a lemma.
Lemma: Let be a group and let . If , then . Proof: Let . By the subgroup criterion (in ) we have . Thus by the subgroup criterion (in ), .
- Let . If , then since is a subgroup of . Thus . Moreover, note that if , we have , so that . Thus . So , and by the lemma we have .
Now consider , , with . We have , so that . Thus .
- Let . Since , we have , and thus . So is abelian. Let and . Since is abelian, we have , so that . Thus .