# Definition:Coset Space

## Definition

Let $G$ be a group, and let $H$ be a subgroup of $G$.

### Left Coset Space

The **left coset space (of $G$ modulo $H$)** is the quotient set of $G$ by left congruence modulo $H$, denoted $G / H^l$.

It is the set of all the left cosets of $H$ in $G$.

### Right Coset Space

The **right coset space (of $G$ modulo $H$)** is the quotient set of $G$ by right congruence modulo $H$, denoted $G / H^r$.

It is the set of all the right cosets of $H$ in $G$.

### Note

If we are (as is usual) concerned at a particular time with only the left or the right coset space, then the superscript is usually dropped and the notation $G / H$ is used for both the left and right coset space.

If, in addition, $H$ is a normal subgroup of $G$, then $G / H^l = G / H^r$ and the notation $G / H$ is then unambiguous anyway.

## Also known as

Some sources call this the **left quotient set** and **right quotient set** respectively.

Some sources use:

- $G \divides H$ for $G / H^l$
- $G / H$ for $G / H^r$

This notation is rarely encountered, and can be a source of confusion.

## Also see

- Definition:Index of Subgroup, the size of a
**coset space**.

## Sources

- 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.2$: Homomorphisms