# Normal Subgroups

Let $G$ be an abelian group, the composition in $G$ being denoted multiplicatively. Let $H$ be any subgroup of $G$. If $x$ is an element of $G$, then $Hx$ is a right coset of $H$ in $G$ and $xH$ is a left coset of $H$ in $G$. Also $G$ is abelian, therefore we must have $Hx = xH\,\,\,\forall x \in G$. However, it is possible that $G$ is not abelian, yet it is possesses a subgroup $H$ such that $Hx = xH\,\,\,\forall x \in G$. Such subgroups of $G$ fall under the category of normal subgroups, and they are very important.

Definition

A subgroup $N$ of a group $G$ is said to be a normal subgroup of $G$ if for every $x \in G$ and for every $n \in N$, $xn{x^{ – 1}} \in N$.
From this definition we can immediately conclude that $N$ is a normal subgroup of $G$ if and only if
$xN{x^{ – 1}} \subset N\,\,\,\,\,\,\,\forall x \in G$