# Conjugation, Normal Subgroups and Factor Groups

## Normal Subgroups

See these notes and this video.

**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

See these notes and this video.

- 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:

is not normal.

## Factor Groups

See these notes and this video.

**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 left 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

See the links above as well as these notes and this video.

- 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 \(\mathbb{Z}_{2}\).
- If \(G=D_{n}\) where \(n\) is even, and \(H=Z(G)\), then \(G/H=D_{n/2}\).
- If \(G=Q\) and \(H=\{-1,1\}\) then \(G/H=\mathbb{Z}_{2}\times\mathbb{Z}_2.\)

## Simple groups

See these notes and this video.

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

**Theorem:** \(A_{n}\) is simple if \(n\ge 5\).

For a proof of this theorem, see Section 10.2 of the text. Other resources include:

- Keith Conrad’s careful proof which uses some terminology we haven’t discussed but probably will.

Two videos:

To get some perspective on this result, there is a long discussion on math stackexchange about it.