# Quotient Groups

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

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

Proof:
Let$N$ be a normal subgroup of a group $G$. Since $N$ is normal in $G$, 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 $G|N$ be the collection of all cosets of $N$ in $G$, i.e. let
$G|N = \left\{ {Na:a \in G} \right\}$

Closure Property: Let $a,b \in G$, then

Since $ab \in G$, therefore $Nab$ is also a coset of $N$ in $G$. So $Nab \in G|N$. Thus $G|N$ is closed with respect to coset multiplication.

Associatively: Let $a,b,c \in G$. Then $Na,Nb,Nc \in G|N$. We have

Thus the product in $G|N$ satisfies the associative law.

Existence of Identity: We have $N = Ne \in G|N$. Also if $Na$ is any element of $G|N$, then

Therefore the coset $N$ is the identity element.

Existence of Inverse: Let $Na \in G|N$, then $N{a^{ - 1}} \in G|N$. We have

Therefore the coset $N{a^{ - 1}}$ is the inverse of $Na$. Thus each element of $G|N$ possesses inverse.

Hence $G|N$ is a group with respect to product of cosets.