Let be an abelian group, the composition in being denoted multiplicatively. Let be any subgroup of . If is an element of , then is a right coset of in and is a left coset of in . Also is abelian, therefore we must have . However, it is possible that is not abelian, yet it is possesses a subgroup such that . Such subgroups of fall under the category of normal subgroups, and they are very important.
A subgroup of a group is said to be a normal subgroup of if for every and for every , .
From this definition we can immediately conclude that is a normal subgroup of if and only if