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:


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