Cosets

Let $H$ be a subgroup of $G$.

Left Cosets

Definition: Left Cosets

Let $a\in G$. The left coset $aH$: $aH=\left\{ah\mid h\in H\right\}$

Example

Consider 4Z, $G=\mathbb{Z}/4\mathbb{Z}=\left\{0,1,2,3\right\}$.

Let the Subgroup $H\subseteq G=\left\{0,2\right\}$.

Proposition 1

Proposition: Any two left cosets of $H$ in $G$ are either identical or disjoint.

In other words, $a,b\in G,aH\cap bH\ne \varnothing ⟹aH=bH$.

Proof: Fix $a,b\in G$. Suppose $aH\cap bH\ne \varnothing$.

There exists an $x\in aH\cap bH$. By definition of $aH$ and $bH$, there exists $h_1\in H$ and $h_2\in H$ such that $a{h}_1=b{h}_2=x$.

$$\begin{array}{rl}a{h}_1=b{h}_2& ⟹a=b{h}_2{h}_1^{-1}\\ & ⟹a\in bH\\ & ⟹aH\subseteq (bH)H=b(HH)=bH.\end{array}$$

Note:

The order of all left cosets of $H$ are the same.

Proposition 2

Proposition: $G$ is the disjoint union of the left cosets of $H$ in $G$.

Right Cosets