Definition:Coset/Left Coset

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


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

Also defined as
The definition given here is the usual one, but some sources (see, for example) order the operands in the opposite direction, and hence $H x$ is a left coset.

Also see

 * Definition:Left Coset Space


 * Definition:Right Coset
 * Definition:Right Coset Space