# Permuting subgroups

From Groupprops

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

## Contents

## Definition

### Definition with symbols

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

- (the product of subgroups) is a subgroup
- Given elements in and in , there exist elements in and in such that . In other words, .
- . In other words, the commutator of and 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