nLab
coset

Definition

For $G$ a group and $H\hookrightarrow G$ a subgroup, the left/right cosets are the $H$-orbits in $G$ under the action by left/right multiplication.

$$G/H=\{gH\mid g\in G\}.$$G/H = \{g H | g \in G\} \,.

Properties

The coset inherits the structure of a group if $H$ is a normal subgroup.

The natural projection $G\to G/H$, mapping the element $g$ to the element $gH$, realizes $G$ as an $H$-principal bundle over $G/H$. We therefore have a homotopy pullback

$$\begin{array}{ccc}G& \to & *\\ \downarrow & & \downarrow \\ G/H& \to & BH\end{array}$$\array{ G & \to&* \\ \downarrow && \downarrow \\ G/H &\to& \mathbf{B}H }

where $BH$ is the delooping groupoid of $H$. By the pasting law for homotopy pullbacks then we get the homotopy pullback

$$\begin{array}{ccc}G/H& \to & BH\\ \downarrow & & \downarrow \\ *& \to & BG\end{array}$$\array{ G/H & \to&\mathbf{B}H \\ \downarrow && \downarrow \\ * &\to& \mathbf{B}G }

Related concepts

• coset space

• coadjoint orbit