Definition:Coset Space/Left Coset Space

Definition

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

The left coset space (or family) of $G$ modulo $H$ is denoted $G / H^l$ and is the set of all the left 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.

Some sources use $G \backslash H$ 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

• Quotient Group

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 6.1$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 2.2$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 37$