# 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 operation 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**.

## Motivation

In Kernel is Normal Subgroup of Domain it was shown that the kernel of a group homomorphism is a normal subgroup of its domain.

In this result it has been shown that *every* normal subgroup is a kernel of at least one group homomorphism of the group of which it is the subgroup.

We see that when a subgroup is normal, its cosets make a group using the group operation defined as in this result.

However, it is not possible to make the left or right cosets of a non-normal subgroup into a group using the same sort of group operation.

Otherwise there would be a group homomorphism with that subgroup as the kernel, and we have seen that this can not be done unless the subgroup is normal.

## 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.

- 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.

## Sources

- 1955: John L. Kelley:
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*Sets and Groups*... (previous) ... (next): $\S 6.7$. Quotient groups - 1965: Seth Warner:
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*A Course in Group Theory*... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Proposition $7.11$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**quotient group** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**quotient group**