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

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

Examples

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.

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.