# 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 $N$**.

## 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 that 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

- Quotient Group is Group, where $G / N$ is proven to be a group

- 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.

## Linguistic Note

The word **quotient** derives from the Latin word meaning **how often**.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 6.7$. Quotient groups - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures - 1966: Richard A. Dean:
*Elements of Abstract Algebra*: $\S 1.10$ - 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): Introduction: Notation - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: Quotient Groups - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.2$: Homomorphisms - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 47$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.7$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 50.3$ Quotient groups - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**quotient group** - 1996: John F. Humphreys:
*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**