Notes on Group Theory

David K. Zhang
Last Modified 2022-03-15

Introduction

Group theory is the first subject in a branch of mathematics known as abstract algebra (or modern algebra). Abstract algebra is one of the main branches of modern pure mathematics, and it is a standard requirement in the university mathematics curriculum. Every professional mathematician and theoretically-oriented physical scientist (i.e., physicists and chemists) should know a thing or two about group theory.

The goal of abstract algebra is to study the general properties of algebraic systems and the interrelations between them. That might not mean a lot if you haven’t studied abstract algebra before, so let me contextualize this statement.

In grade school, you learned about a handful of different number systems, including the integers $\Z$ , the rational numbers $\Q$ , the real numbers $\R$ , and possibly (if you were an advanced student) the complex numbers $\C$ . You also learned that there are several algebraic properties that these systems satisfy, such as the commutative property of addition, $x + y = y + x$ , and the associative property of multiplication, $x \cdot (y \cdot z) = (x \cdot y) \cdot z$ .

In abstract algebra, we ask what other number systems support a notion of addition, negation, multiplication, or some other operation, that satisfies the commutative, associative, distributive, etc. property.

Each of the number systems mentioned above supports the four basic arithmetic operations of addition, subtraction, multiplication, and (with the exception of the integers) division. Of course, subtraction and division are merely the inverses of addition and multiplication, so in each case, there are really only two fundamental arithmetic operations. [TODO: Finish writing introduction.]

Definition: Group

A group is an algebraic structure $\alg{G; 1, {}^{-1}, \cdot}$ consisting of:

a set $G$ , called the underlying set;
a distinguished element $1 \in G$ , called the identity element;
a unary operation ${}^{-1}: G \to G$ , written as $x \mapsto x^{-1}$ , called inversion;
a binary operation $\cdot : G \times G \to G$ , written as $(x, y) \mapsto x \cdot y$ , called the group operation or group product;

satisfying the following requirements:

Associative property: $(x \cdot y) \cdot z = x \cdot (y \cdot z)$ for all $x, y, z \in G$ .
Identity property: $1 \cdot x = x \cdot 1 = x$ for all $x \in G$ .
Inverse property: $x \cdot x^{-1} = x^{-1} \cdot x = 1$ for all $x \in G$ .

Definition: Subgroup

TODO

Definition: Abelian Group

An abelian group is a group $\alg{G; 1, {}^{-1}, \cdot}$ that satisfies the following additional requirement:

Commutative property: $x \cdot y = y \cdot x$ for all $x, y \in G$ .

Definition: Left Coset, Right Coset, $G/H$ , $H \backslash G$

Let $G$ be a group, and let $H \le G$ be a subgroup. A left coset of $H$ is a set of the form

gH \coloneqq \{ gh : h \in H \}

for some fixed element $g \in G$ . Similarly, a right coset of $H$ is a set of the form

Hg \coloneqq \{ hg : h \in H \}

for some fixed element $g \in G$ . The collection of all left cosets of $H$ is denoted by $G/H$ , while the collection of all right cosets of $H$ is denoted by $H \backslash G$ .

This definition introduces a new notational convention. Whenever we apply a group operation, such as $gH$ or $H^{-1}$ , to a subset of a group, we mean the set formed by applying that operation to each element of the subset. For example:

\begin{aligned} gH &\coloneqq \{ gh : h \in H \} \\ H^{-1} &\coloneqq \{ h^{-1} : h \in H \} \\ g_1 H g_2 &\coloneqq \{ g_1 h g_2 : h \in H \} \end{aligned}

Note that we can write $g_1 H g_2$ without ambiguity, since the associative property guarantees that $(g_1 H) g_2 = g_1 (H g_2)$ .

Definition: Index, $\abs{G:H}$

Let $G$ be a group, and let $H \le G$ be a subgroup. The index of $H$ in $G$ , denoted by $\abs{G:H}$ , is the cardinality of the

Cosets Partition a Group

Theorem: Let $G$ be a group, and let $H \le G$ be a subgroup.

Proof:

Suppose two cosets $xH$ and $yH$ of $H$ intersect, i.e., there exists an element $z \in xH \cap yH$

Definition: Normal Subgroup

Let $G$ be a group. We say that a subgroup $H \le G$ is normal if for all $g \in G$ and $h \in H$ , we have $g^{-1}hg \in H$ .

left conjugation — this is a left action

{}^g x \coloneqq g x g^{-1}

{}^h ({}^g x) = {}^h (g x g^{-1}) = h (g x g^{-1}) h^{-1} = (hg) x (hg)^{-1} = {}^{hg} x

right conjugation — this is a right action

x^g \coloneqq g^{-1} x g

(x^g)^h = (g^{-1} x g)^h = h^{-1} (g^{-1} x g) h = (gh)^{-1} x (gh) = x^{gh}

Definition: Central Element, Center, $Z(G)$

Let $G$ be a group. An element $g \in G$ is central if $g$ commutes with every element of $G$ . The set of all central elements of $G$ is called the center of $G$ , and is denoted by $Z(G)$ .

Z(G) \coloneqq \{ g \in G : \forall h \in G,\ gh = hg \}

Center is a Subgroup

Theorem: Let $G$ be a group. Its center $Z(G)$ is a subgroup of $G$ .

Proof: We must show that $Z(G)$ contains the identity element $1$ and is closed under taking inverses and products.

Clearly, $1 \in Z(G)$ , since the identity element $1$ commutes with everything.
Let $a \in Z(G)$ and $g \in G$ . By definition, we have $ag = ga$ . By multiplying on the left and right by $a^{-1}$ , we obtain $ga^{-1} = a^{-1}g$ . Hence, $a^{-1}$ commutes with $g$ . Since $g \in G$ was arbitrary, this proves that $a^{-1} \in Z(G)$ .
Let $a, b \in Z(G)$ and $g \in G$ . Because $a$ and $b$ are central, we can write $abg = agb = gab$ . Hence, $ab$ commutes with $g$ . Again, because $g \in G$ was arbitrary, this proves that $ab \in Z(G)$ . □

Center is a Normal Subgroup

Theorem: Let $G$ be a group. Its center $Z(G)$ is a normal subgroup of $G$ .

Proof: Let $g \in G$ and $z \in Z(G)$ . Because $z$ is central, we have $g^{-1}zg = zg^{-1}g = z \in Z(G)$ . □

Center is a Normal Subgroup

Theorem: Let $G$ be a group. Its center $Z(G)$ is a normal subgroup of $G$ .

Proof: Let $g \in G$ and $z \in Z(G)$ . Because $z$ is central, we have $g^{-1}zg = zg^{-1}g = z \in Z(G)$ . □