Permuting subgroups

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This article defines a symmetric relation on the collection of subgroups inside the same group.

Definition

Definition with symbols

Two subgroups H and K of a group G are termed permuting subgroups if the following equivalent conditions hold:

  1. HK = KH
  2. HK (the product of subgroups) is a subgroup
  3. Given elements h in H and k in K, there exist elements k' in K and h' in H such that hk = k'h'. In other words, HK \subseteq KH.
  4. [H,K] \subseteq HK. In other words, the commutator of H and K is contained in their product.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of permuting subgroups

Relation with other relations

Stronger relations

Weaker relations