Definition:Coset Space

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

Left Coset Space
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 $$G$$.

Right Coset Space
The right coset space (or family) of $$G$$ modulo $$H$$ is denoted $$G / H^r$$ and 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.

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

Some sources use:
 * $$G \backslash H$$ for $$G / H^l$$
 * $$G / H$$ for $$G / H^r$$

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