Subgroup/Normal subgroup/Definition

Normal subgroup

Let ${}G$ be a group, and let ${}H\subseteq G$ denote a subgroup. ${}H$ is called a normal subgroup if

${}xH=Hx\,$

holds for all ${}x\in G$, that is, if every left coset of ${}x$ coincides with the right coset of ${}x$.