Coset

algebra

Definition

Nebenklasse

Coset

A coset is a subset formed by multiplying all elements of a subgroup by a fixed group element.

Let $G$ be a group and $H\le G$ a subgroup of $G$, for an element $g\in G$:

1. The left coset (“Linksnebenklasse”) of $H$ with respect to $g$ is:

$$g\cdot H=\{g\cdot h\mid h\in H\}$$

2. The right coset (“Rechtsnebenklasse”) of $H$ with respect to $g$ is:

$$H\cdot g=\{h\cdot g\mid h\in H\}$$