# Notes on Group Theory

Last Modified 2022-03-15

# Introduction

Group theory is the first subject in a branch of mathematics known as * abstract algebra* (or

*). 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.*

**modern algebra**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 * algebraic properties* that these systems satisfy, such as the

*of addition,*

**commutative property***of multiplication,*

**associative property**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

- 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 theor**group operation**;**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: Abelian Group**

An * abelian group* is a group

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