Normal Subgroups

Introduction

The set of cosets of a subgroup raises a natural question: can this set itself be endowed with a group structure? This leads to the concept of normal subgroups, which are essential for constructing quotient groups.

Definition

Definition 2.3: A subgroup $N$ of a group $G$ is a normal subgroup if for every $g \in G$ , $g N g^{- 1} = N$ . This is denoted by $N ⊴ G$ .

The condition $g N g^{- 1} = N$ is equivalent to $g N = N g$ for all $g \in G$ .

Equivalent Characterizations

A subgroup $N$ of $G$ is normal if and only if any of the following conditions hold:

Conjugation condition: $g N g^{- 1} = N$ for all $g \in G$
Coset condition: $g N = N g$ for all $g \in G$
Kernel condition: $N$ is the kernel of some homomorphism from $G$ to another group
Invariance condition: $N$ is invariant under all inner automorphisms of $G$

Examples

Example 1: Normal Subgroups in Abelian Groups

In an abelian group, every subgroup is normal because $g n g^{- 1} = n$ for all $g, n \in G$ .

Examples:

In $(Z, +)$ , every subgroup $n Z$ is normal
In $(Q^{*}, \cdot)$ , the subgroup ${1, - 1}$ is normal

Example 2: Normal Subgroups in $D_{4}$

Consider the subgroup $N = {e, r^{2}}$ in $D_{4}$ :

For any rotation $r^{k}$ : $r^{k} N = N r^{k}$ (since $r^{2}$ commutes with rotations)
For any reflection $s r^{k}$ : $(s r^{k}) N = N (s r^{k})$ (since $r^{2}$ is in the center)

Therefore, $N$ is normal, and $D_{4} / N$ is a group of order 4.

Example 3: The Alternating Group

The alternating group $A_{n}$ is a normal subgroup of the symmetric group $S_{n}$ because:

$A_{n}$ is the kernel of the sign homomorphism $sgn : S_{n} \to {\pm 1}$
The quotient group $S_{n} / A_{n}$ is isomorphic to ${\pm 1}$

Example 4: Center of a Group

The center $Z (G)$ of a group $G$ is always a normal subgroup because:

$Z (G) = {z \in G ∣ g z = z g for all g \in G}$
For any $g \in G$ and $z \in Z (G)$ : $g z g^{- 1} = z \in Z (G)$

Properties

Intersection of Normal Subgroups

The intersection of any collection of normal subgroups is itself a normal subgroup.

Product of Normal Subgroups

If $N$ and $M$ are normal subgroups of $G$ , then $N M = M N$ is also a normal subgroup of $G$ .

Subgroup of Index 2

Any subgroup of index 2 is normal.

Proof: If $[G : H] = 2$ , then there are exactly two left cosets: $H$ and $g H$ for some $g \notin H$ . Similarly, there are exactly two right cosets: $H$ and $H g^{'}$ for some $g^{'} \notin H$ . Since $g H \cup H = G = H \cup H g^{'}$ , we must have $g H = H g^{'}$ , which implies $H$ is normal.

Non-Normal Subgroups

Example 1: Subgroups in $S_{3}$

Consider the subgroup $H = {e, (1, 2)}$ in $S_{3}$ :

$(1, 3) H = {(1, 3), (1, 2, 3)}$
$H (1, 3) = {(1, 3), (1, 3, 2)}$
Since $(1, 3) H \neq H (1, 3)$ , $H$ is not normal

Example 2: Subgroups in $D_{4}$

Consider the subgroup $H = {e, s}$ in $D_{4}$ :

$r H = {r, r s} = {r, s r^{3}}$
$H r = {r, s r} = {r, s r}$
Since $r H \neq H r$ , $H$ is not normal

Applications

Application 1: Quotient Groups

Normal subgroups are essential for constructing quotient groups, which are fundamental in group theory.

Application 2: Homomorphisms

Normal subgroups are precisely the kernels of group homomorphisms.

Application 3: Group Structure

Normal subgroups help us understand the structure of groups through the isomorphism theorems.