Permuting subgroups

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 $k^{\prime }$ in $K$ and $h^{\prime }$ in $H$ such that $hk=k^{\prime }{h}^{\prime }$. 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

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

Weaker relations

• Elliptic pair of subgroups