# Permutable complements

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

## Contents

## Definition

### Symbol-free definition

Two subgroups of a group are said to be permutable complements if:

- Their intersection is trivial
- Their product is the whole group

### Definition with symbols

Two subgroups and of a group are termed **permutable complements** if the following two conditions hold:

- is the trivial group

## Facts

### Permutable complements need not be unique

Given a subgroup of , there may or may not exist permutable complements of . Moreover, there may exist multiple possibilities for a complement to , and the multiple possibilities might not be pairwise isomorphic.

`Further information: Every group of given order is a permutable complement for symmetric groups, Retract not implies normal complements are isomorphic`

### For a normal subgroup, they are fixed upto isomorphism

Interestingly, when a subgroup is normal, then any two permutable complements to it must be isomorphic. In fact, any permutable complement to it must be isomorphic to the quotient group.

- There may be multiple subgroups that are pairwise permutable complements
- Retract not implies every permutable complement is normal
- Permutable complement to normal subgroup is isomorphic to quotient
- Permutable complements to abelian normal subgroup are automorphic
- Retract not implies normal complements are isomorphic
- Permutable complements to normal subgroup need not be automorphic