# 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 $k'$ 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