Definition:Coset

Definition

Let $G$ be a group, and let $H \le G$.

Left Coset

The left coset of $x$ modulo $H$, or left coset of $H$ by $x$, is:

$x H = \left\{{y \in G: \exists h \in H: y = x h}\right\}$

This is the equivalence class defined by left congruence modulo $H$.

Alternatively, it can be viewed as an extension of the idea of the subset product:

$x H = \left\{{x}\right\} H$

Right Coset

The right coset of $y$ modulo $H$, or right coset of $H$ by $y$, is:

$H y = \left\{{x \in G: \exists h \in H: x = h y}\right\}$

This is the equivalence class defined by right congruence modulo $H$.

Alternatively, it can be viewed as an extension of the idea of the subset product:

$H y = H \left\{{y}\right\}$

Also see

• Coset Space

• Results about cosets can be found here.

Sources

• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.9$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 5.20$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 2.2$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 41$
• John F. Humphreys: A Course in Group Theory (1996): $\S 5$: Definition $5.1$