Given a subgroup of group $G$ , the left cosets of $H$ in $G$ are sets of the form ${g h : h \in H}$ , for some $g \in G$ . This is written $g H$ as a shorthand.

Similarly, the right cosets are the sets of the form $H g = {h g : h \in H}$ .

Examples

Symmetric group

In $S_{3}$ , the symmetric group on three elements, we can list the elements as ${e, (123), (132), (12), (13), (23)}$ , using cycle notation. Define $A_{3}$ (which happens to have a name: the alternating group) to be the subgroup with elements ${e, (123), (132)}$ .

Then the coset $(12) A_{3}$ has elements ${(12), (12) (123), (12) (132)}$ , which is simplified to ${(12), (23), (13)}$ .

The coset $(123) A_{3}$ is simply $A_{3}$ , because $A_{3}$ is a subgroup so is closed under the group operation. $(123)$ is already in $A_{3}$ .

more examples, with different requirements

Properties

The left cosets of $H$ in $G$ partition $G$ . (Proof.)

For any pair of left cosets of $H$ , 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 $H$ 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.

there must be a less clumsy way to do it

Additionally, there is a key theorem whose usual proof considers cosets (Lagrange's theorem) which strongly restricts the possible sizes of subgroups of $G$ , and which itself is enough to classify all the groups of order $p$ for $p$ prime. Lagrange's theorem also has very common applications in number_theory, in the form of the Fermat-Euler theorem.