Definition:Coset

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

Left Coset
The left coset of $$x$$ modulo $$H$$ 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 $$H$$ 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:

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