# Conjugation, Normal Subgroups and Factor Groups

## Normal Subgroups

Definition: A subgroup $H$ of a group $G$ is normal if the left cosets and right cosets of $H$ are the same. That is, $H$ is normal if, for all $g\in G$, $gH=Hg$.

### Examples

• Every subgroup of an abelian group is normal.
• The subgroup of rotations of an n-gon is normal in the Dihedral group $D_{n}$.
• The group $\mathrm{SL}_{2}(\mathbb{R})$ of $2\times 2$ matrices with real entries and determinant one is normal in $\mathrm{GL}_{2}(\mathbb{R})$.
• The subgroup $A_{n}$ of even permutations is normal in the symmetric group $S_{n}$.
• The subgroup $\{-1,1\}$ is normal in the quaternion group $Q=\{\pm 1, \pm i, \pm j, \pm k\}.$
• If $G$ is a group, the center $Z(G)$ is normal in $G$.

### Non-examples

• The subgroup of $D_{n}$ generated by a reflection $s$ is not a normal subgroup.
• The subgroup of $S_{n}$ generated by a cycle is not a normal subgroup.
• The subgroup $H$ of $\mathrm{GL}_{2}(\mathbb{R})$ consisting of matrices of the form:
$H=\{\left(\begin{matrix} 1 & x \\ 0 & 1 \end{matrix}\right) : x\in\mathbb{R}\}$

is not normal.

## Factor Groups

Definition: Let $G$ be a group and $H$ be a normal subgroup. Let $G/H$ be the set of left cosets of $H$ in $G$. Introduce a multiplication on cosets by the rule $(aH)(bH) = abH.$

Proposition: $G/H$, with the operation described above, is a group, called the “quotient group” or “factor group” of $G$ by $H$.

Key points in the proof:

• The operation is well defined, meaning that if $aH=a’H$ and $bH=b’H$ then $abH=a’b’H$. This is where the normal hypothesis is needed.
• The coset $H$ is the identity element.
• The inverse of $aH$ is $a^{-1}H$.

Note that $G/H$ has $[G:H]=\frac{\vert G\vert}{\vert H\vert}$ elements.

### Examples

• If $G=\mathbb{Z}$ and $H=n\mathbb{Z}$ then $G/H=\mathbb{Z}_{n}$.
• If $G=\mathbb{Z}_{n}$ and $H=d\mathbb{Z}_{n}$ where $d$ is a divisor of $n$ then $G/H=\mathbb{Z}_{d}.$
• If $G=\mathrm{GL}_{2}(\mathbb{R})$ and $H=\mathrm{SL}_{2}(\mathbb{R})$ then $G/H=\mathbb{R}^{\star}$.
• If $G=D_{4}$ and $H=Z(G)$ then $G/H$ is isomorphic to $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$.
• If $G=D_{n}$ and $H$ is the subgroup of rotations, then $G/H$ is isomorphic to $\mathbb{Z}_{2}$.

## Simple groups

Definition: A group $G$ is simple if it has no nontrivial normal subgroups.

Theorem: $A_{n}$ is simple if and only if $n\ge 5$.