# Theorems of Normal Subgroups

__Theorem 1__**:** A subgroup of a group is normal if and only if .

__Proof__**:** Let , then . Therefore is a normal subgroup of .

Conversely, let be a normal subgroup of . Then

Also . Therefore we have

From (i) and (ii) we can conclude that

__Theorem 2__**: **A subgroup of a group is a normal subgroup of if and only if each left coset of in is a right coset of in .

__Proof__**: **Let be a normal subgroup of then

each left coset is the coset

Conversely, let each left coset of in be a right coset of in . This means that if is any element of , then the left coset is also a right coset. Now , and therefore . So must also belong to that right coset which is equal to left coset . But is an element of the right coset , and two right cosets are either disjointed or identical, i.e. if two right cosets contain one common element then they are identical. Therefore is the unique right coset which is equal to the left coset .

Therefore, we have

is normal a subgroup of .

__Theorem 3__**: **A subgroup of a group is a normal subgroup of if and only if the product of two right cosets of in is again a right coset of in .

__Theorem 4__**:** The intersection of two normal subgroups of a group is a normal subgroup.** **