# Definition:Coset Space/Left Coset Space

## Definition

Let $G$ be a group.

Let $H$ be a subgroup of $G$.

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 $G$.

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## Also known as

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

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

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

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

## Notation

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.

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.

## Also see

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 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 - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 37$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions