Definition:Coset

Definition
Let $G$ be a group, and let $H \le G$.

Left Coset
The left coset of $x$ modulo $H$, or left coset of $H$ by $x$, is:


 * $x H = \left\{{y \in G: \exists h \in H: y = x h}\right\}$

This is the equivalence class defined by left congruence modulo $H$.

Alternatively it can be viewed as an extension of the idea of the subset product:


 * $x H = \left\{{x}\right\} H$

Right Coset
Similarly, the right coset of $y$ modulo $H$, or right coset of $H$ by $y$, is:


 * $H y = \left\{{x \in G: \exists h \in H: x = h y}\right\}$

This is the equivalence class defined by right congruence modulo $H$.

Again, it can be viewed as an extension of the idea of the subset product:


 * $H y = H \left\{{y}\right\}$