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 come under the category of normal subgroups and these are very important.

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