Product of subgroups
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 :
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.