# Quotient Groups

__Definition__**:** If is a group and is a normal subgroup of group , then the set of all cosets of in is a group with respect to multiplication of cosets. It is called the quotient group or factor group of by . The identity element of the quotient group by .

__Theorem__**:** The set of all cosets of a normal subgroup is a group with respect to multiplication of complexes as the composition.

__Proof__**:**

Let be a normal subgroup of a group . Since is normal in , therefore each right coset will be equal to the corresponding left coset.

Thus there is no distinction between right and left cosets and we shall call them simply as cosets. Let be the collection of all cosets of in , i.e. let

**Closure Property: **Let , then

Since , therefore is also a coset of in . So . Thus is closed with respect to coset multiplication.

**Associatively:** Let . Then . We have

Thus the product in satisfies the associative law.

**Existence of Identity: **We have . Also if is any element of , then

Therefore the coset is the identity element.

**Existence of Inverse:** Let , then . We have

Therefore the coset is the inverse of . Thus each element of possesses inverse.

Hence is a group with respect to product of cosets.