Definition:Quotient Group

Definition
Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.

Then the left coset space $G / N$ is a group, where the group product is defined as:
 * $\left({a N}\right) \left({b N}\right) = \left({a b}\right) N$

$G / N$ is called the quotient group of $G$ by $N$.

It is proven to be a group in Quotient Group is Group.

Also known as
A quotient group is also known as a factor group.

Also see

 * From Subgroup is Normal iff Left Cosets are Right Cosets, the left coset space is equal to the right coset space.

It follows that $G / N$ does not depend on whether left cosets are used to define it or right cosets.

Thus we do not need to distinguish between the left quotient group and the right quotient group - the two are one and the same.