# Equivalence of definitions of permuting subgroups

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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)