If is a group, is a subgroup and be any element in then the set
is called the right coset generated by and is denoted by .
Similarly, the set
is called left coset generated by and is denoted by .
Since , we see that itself is a right as well as left coset. Moreover, since , it is evident that .
If the group operation is addition, we define the right coset of in by
Similarly, left coset of in is defined by
It must be noticed that cosets are not necessarily subgroups of . They are only special types of complexes which sometimes called residue classes modulo subgroup.
In general . In 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 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 with and is identical with .