# Definition:Coset Space/Right Coset Space

## Definition

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

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 $G$.

## Also known as

Some sources call this the **right quotient set**.

Others call it the **right coset family**.

Others use the term **right coset decomposition**.

Some sources use a hyphen: **right-coset space** or **right-coset family**, and so on.

Some sources use $G \divides H$ for the left coset space, reserving $G / H$ for the right coset space.

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

If we are (as is usual) concerned at a particular time with only the right or the left coset space, then the superscript is usually dropped.

Thus the notation $G / H$ is used for both the right and left 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 see

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions