Coset

definitiongroup-theory

Definition

For a group $(G,\cdot )$, subgroup $H\le G$, and element $g\in G$, we define the left coset of $g$ with respect to $H$ in $G$, as,

$$gH=\{gh\mid h\in H\},$$

and we define the right coset of $g$ with respect to $H$ in $G$, as,

$$Hg=\{hg\mid h\in H\}.$$

Properties

1. Distinct left cosets are disjoint, as are distinct right cosets.
2. Distinct left cosets in $G$ partition $G$.
3. If $H$ is finite, then $\mid H\mid =\mid gH\mid =\mid Hg\mid$.

Examples