# 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:

$x H = \set {y \in G: \exists h \in H: y = x h}$

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

That is, it is the subset product with singleton:

$x H = \set x H$

## Also defined as

The definition given here is the usual one, but some sources (see P.M. Cohn: Algebra Volume 1 (2nd ed.), for example) order the operands in the opposite direction, and hence $H x$ is a left coset.

## Also see

• Results about cosets can be found here.