Definition:Coset Space
Contents
Definition
Let $G$ be a group, and let $H$ be a subgroup of $G$.
Left Coset Space
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 $G$.
Right Coset Space
The right coset space (of $G$ modulo $H$) is the quotient set of $G$ by right congruence modulo $H$, denoted $G / H^r$.
It 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.
Also known as
Some sources call this the left quotient set and right quotient set respectively.
Some sources use:
- $G \divides H$ for $G / H^l$
- $G / H$ for $G / H^r$
This notation is rarely encountered, and can be a source of confusion.
Also see
- Definition:Index of Subgroup, the size of a coset space.
Sources
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms
