Definition – Left Coset – Math-reference.net

Contents
1. Definitions
2. Theorems

Definition – Left coset

Let $\langle G,*\rangle$ be a group, and let $H\subset G$ be a subgroup of $G$ . We define a left coset of $H$ in $G$ to be the set

$gH = \{g*h ~|~ h\in H\}$

For any element $g\in G$ .

Theorems

Theorem 1 – $s\in gH \implies sH=gH$

Proof –

From the definition of the left coset, we have

Theorem 2 –

The family of all cosets of $H$ in $G$ as $g$ ranges over $G$ , partitions the set $H$ .

Proof

We must prove that the different cosets of the form $gH$ in $G$ are disjoint, or equal. Let us first assume that some element $x$ is shared between two cosets, ie,

Theorem 3 – Let $H$ be a normal subgroup of a group $G$ . Let $aH = cH$ and $bH = dH$ be left cosets of $H$ in $G$ . Then, $(ab)H = (cd)H$ .

Proof

First considering $aH = cH$ , we have that for some $h_1\in H$ , $a = h_1 c$ . And for $bH = dH$ , we have that $b=h_2 d$ for some $h_2\in H$ .

Now we make use of the first part of the antecedent, that $H$ is a normal subgroup of $G$ . As $ghg^{-1}\in H$ for all $g\in G, h\in H$ , we consider the case where $h=h_1,~g = d^{-1}$ . This gives $d^{-1} h_1 d = h_3 \in H$ . Thus, $h_1 d = d h_3$ (where possibly $h_1=h_3$ ), and so

$ab = (ch_1)(dh_2) = c(h_1 d) h_2 = c(d h_3) h_2 = (cd)(h_3 h_2) = (cd)h_4$ .

But $(cd)h_4 = ab \in (cd)H$ , which implies that $(cd)H = (ab)H$ . $\square$ .