Definition:Coset Space/Left Coset Space

Definition
Let $G$ be a group, and let $H$ be a subgroup of $G$. The left coset space (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.

Others call it the left coset family.

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.

Also see

 * Definition:Quotient Group


 * Definition:Right Coset Space