Quotient Group

Let $H$ be a subgroup of $G$. A left coset of $H$ corresponding to $g$ is given by

$$gH=\{gh:h\in H\}$$

where as a right coset of $H$ corresponding to $g$ is given by

$$Hg=\{hg:h\in H\}.$$

Notice that you may have some $g$ and $g^{\prime }$ such that $gH=g^{\prime }H$. In such a case, $g$ and $g^{\prime }$ are both representatives of the same coset and we say $g\sim g^{\prime }$.

Whether or not the left and right cosets coincide depend on the underlying subgroup.

Normal

A subgroup $H\le G$ is normal if for all $g\in G$

$$gH=Hg.$$

Normal subgroups are denoted by $H⊴G$.

Note that if the group $G$ is abelian, then every subgroup is normal since

$$g+H=\{g+h:h\in H\}=\{h+g:h\in H\}=H+g.$$

Define $G/H$ to be all left cosets of $H$, i.e.,

$$G/H=\{gH:g\in G\}.$$

We define the operation $(aH)(bH)=(ab)H$. In order for $G/H$ to be a group, we need to ensure $(aH)(bH)=(a^{\prime }H)(b^{\prime }H)$ for any representatives $a\sim a^{\prime }$ and $b\sim b^{\prime }$.

Thm

If $H⊴G$, then $G/H$ is a group.

\begin{proof} Suppose that $xH=aH$ and $yH=bH$. Notice that

$$(ab)H=a(bH)=a(yH)=a(Hy)=(aH)y=(xH)y=x(Hy)=x(yH)=(xy)H.$$

Therefore, $(ab)H=(xy)H$.\end{proof}

We call the cardinality of $G/H$ the index and denote it by $[G:H]$. The index is a count of the number of (left) cosets.

Thm

If $G$ is a finite group and $H⊴G$, then the order of $G/H$ is given by

$$|G/H|=|G|/|H|.$$