Group theory/Left and right coset/Definition

coset

Let ${}G$ be a group, and let ${}H\subseteq G$ be a subgroup. For every ${}x\in G$, the subset

${}xH={\left\{xh\mid h\in H\right\}}\,$

is called the left coset of ${}x$ in ${}G$ with respect to ${}H$. Every subset of this form is called a left coset. Accordingly, a set of the form

${}Hx={\left\{hx\mid h\in H\right\}}\,$

is called the right coset (of ${}x$).