If is a group, is a subgroup and is any element in , then the set
is called the right coset generated by and is denoted by .
Similarly, the set
is called the left coset generated by and is denoted by .
Since , we see that itself is a right as well as a left coset. Moreover, since , it is evident that .
If the group operation is addition, we define the right coset of in by
Similarly, the left coset of in is defined by
It must be noted that cosets are not necessarily subgroups of . Rather, they are only special types of complexes which are sometimes called residue classes modulo subgroup.
In general . In the case of an abelian group, each right coset coincides with the corresponding left coset.
Example: Let , and , is a group of . Find all the cosets of in and prove that is equal to the union of all these cosets and also establish that any two cosets are either disjoint or identical.
Solution: We have
Thus there are only distinct cosets, namely and which is disjoint. Also is identical to and is identical to .