Permuting subgroups

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 defining ingredient::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.

Stronger relations

 * One is a normalizing subgroup for the other
 * Mutually permuting subgroups
 * Totally permuting subgroups
 * Conjugate-permuting subgroups

Weaker relations

 * Elliptic pair of subgroups