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 .
Examples
Symmetric group
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 .
Properties
For any pair of left cosets of , there is a bijection between them; that is, all the cosets are all the same size. (Proof.)
Why are we interested in cosets?
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.