# Product of subgroups

From Groupprops

## Definition

### Symbol-free definition

The **Product** of two subgroups of a group is the subset consiting of the paarwise products between the two subgroups.

### Definition with symbols

The **Product** of two subgroups and of a group is:

.

If is abelian and if the group operation is denoted as , the *product* is termed the *sum*, and is denoted :

.

## Facts

is the double coset , being the identity element of .

The cardinality of is .

The product is in general not a subgroup, because it may not be closed under the group operation.

The smallest subgroup containing is the join of and , which is also the subgroup generated by and .

Following statements are equivalent:

- is a subgroup
- , viz., it is precisely the join of and (the subgroup generated by and )

If the above equivalent conditions hold, and are termed permuting subgroups.