Equivalence of definitions of permuting subgroups

Statement
Suppose $$G$$ is a group, and $$H,K$$ are subgroups of $$G$$. Then, the following are equivalent:


 * 1) $$HK$$ (the product of subgroups)is a subgroup
 * 2) $$HK = KH$$
 * 3) $$HK = \langle H, K \rangle$$
 * 4) $$HK \subseteq KH$$
 * 5) $$KH \subseteq HK$$

Textbook references

 * , Page 75, Exercise 7(a) of Section 8 (Products of Groups)