Given a subgroup of group , the left cosets of in are sets of the form , for some . This is written as a shorthand.
Similarly, the right cosets are the sets of the form .
In , the symmetric group on three elements, we can list the elements as , using cycle notation. Define (which happens to have a name: the alternating group) to be the subgroup with elements .
Then the coset has elements , which is simplified to .
The coset is simply , because is a subgroup so is closed under the group operation. is already in .
Under certain conditions (namely that the subgroup must be normal), we may define the quotient_group, a very important concept; see the page on "left cosets partition the parent group" for a glance at why this is useful.
Additionally, there is a key theorem whose usual proof considers cosets (Lagrange's theorem) which strongly restricts the possible sizes of subgroups of , and which itself is enough to classify all the groups of order for prime. Lagrange's theorem also has very common applications in number_theory, in the form of the Fermat-Euler theorem.