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
View a complete list of pages giving proofs of equivalence of definitions

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)