Definition:Coset/Right Coset

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

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


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

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 $x H$ is a right coset.

Also see

 * Definition:Right Coset Space


 * Definition:Left Coset
 * Definition:Left Coset Space