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 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 right or the left coset space, then the superscript is usually dropped and 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 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 $G \mathop \backslash H$ 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.