Normal Groups

Overview

A subgroup $H$ of a group $G$ is normal if its left and right cosets coincide. This is usually denoted as relation $H⊴G$ or $H⊲G$ (if the subset relation is strict). Each of the following characterizations are equivalent:

1. $gh{g}^{-1}\in H$ for all $g\in G$ and $h\in H$.
2. $gH{g}^{-1}=H$ for all $g\in G$.
3. $gH=Hg$ for all $g\in G$.

The second criterion above is often taken as the definition of a normal subgroup.

Subgroups

Both the improper subgroup and trivial subgroup of $G$ are normal.

Product Sets

Let $G$ be a group and $H_1,H_2,\dots ,H_n⊴G$. Then their product set $\prod H_i$ is a normal subgroup of $G$ and is the smallest such subgroup containing each of $H_1,H_2,\dots ,H_n$.

Intersections

The intersection of a collection of normal subgroups is also a normal subgroup.

Quotient Groups

Let $H$ be a normal subgroup of group $G$. Then the cosets of $H$ form the quotient group (or factor group) of $G$ by $H$, denoted $G/H$, under the binary operation

Properties of the subgroup can transmit new properties to the quotient group. For example:

• If $H$ is a normal subgroup of group $G$, then $G/H$ is abelian.
• If $T$ is the torsion subgroup of an abelian group $G$, then $G/T$ is torsion-free.
• If $C$ is the commutator subgroup of group $G$, then $G/C$ is an abelian group.
• If $G$ is a cyclic group and $H⊴G$, then $G/H$ is cyclic.

If $G=H\otimes K$ is a direct product of groups $H$ and $K$, then $\bar{K}=\{(e,k)\mid k\in K\}$ is a normal subgroup of $G$. Also $G/\bar{K}\simeq H$.

Canonical Homomorphism

Every quotient group $G/H$ gives rise to a homomorphism $\gamma :G\to G/H$ given by $\gamma (a)=aH$ and satisfying $\text{Ker}(\gamma )=H$. If $\varphi :G\to G^{\prime }$ is a group homomorphism with $H=\text{Ker}(\varphi )$, then $\gamma :G\to G/H$ is called the canonical homomorphism of $\varphi$.

Simple Groups

A group is simple if it is nontrivial and has no proper nontrivial normal subgroups.

A maximal normal subgroup of a group $G$ is a proper normal subgroup $M$ such that there is no proper normal subgroup $N$ of $G$ properly containing $M$.

$M$ is a maximal normal subgroup of $G$ if and only if $G/M$ is simple.

Centers

Let $G$ be a group. Then the center of $G$ is defined as

$Z(G)$ is an abelian normal subgroup of $G$. If $Z(G)=\{e\}$, we say the center of $G$ is trivial. A group $G$ is abelian if and only if $Z(G)=G$.

Commutator Subgroups

Let $G$ be a group. An element of $G$ that can be expressed in the form $ab{a}^{-1}{b}^{-1}$ for some $a,b\in G$ is a commutator in $G$. Such an element is denoted as $[a,b]$.

Given subgroups $H,K\le G$, we denote the subgroup generated by the commutators $[h,k]$ for all $h\in H$ and $k\in K$ as

The set of all commutators of $G$ generates a normal subgroup $C$ of $G$. This group $C$ is called the commutator subgroup, alternatively denoted as $[G,G]$.

If $N$ is a normal subgroup of $G$, then $G/N$ is abelian if and only if $C\le N$.