Equivalence of definitions of permuting subgroups

This article gives a proof/explanation of the equivalence of multiple definitions for the term permuting subgroups
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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. $KH\subseteq HK$

References

Textbook references

• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 75, Exercise 7(a) of Section 8 (Products of Groups)