Definition:Coset Space/Left Coset Space
Definition
Let $G$ be a group.
Let $H$ be a subgroup of $G$.
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 $G$.
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Also known as
Some sources call this the left quotient set.
Others call it the left coset family.
Others use the term left coset decomposition.
Some sources use a hyphen: left-coset space or left-coset family, and so on.
Notation
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.
Thus 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.
Some sources use $G \divides H$ 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
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 37$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions