This article defines a symmetric relation on the collection of subgroups inside the same group.
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
- One is a normalizing subgroup for the other
- Mutually permuting subgroups
- Totally permuting subgroups
- Conjugate-permuting subgroups