# 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:

$\paren {a N} \paren {b N} = \paren {a b} N$

$G / N$ 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

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.

• Results about quotient groups can be found here.

## Historical Note

The idea of a quotient group appeared in the work of Marie Ennemond Camille Jordan in the $1860$s.

However, the modern formulation using cosets did not appear till the work of Otto Ludwig Hölder in $1889$, fairly late on in the history of group theory.