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\Z, the rational numbers Q\Q, the real numbers R\R, and possibly (if you were an advanced student) the complex numbers C\C. You also learned that there are several algebraic properties that these systems satisfy, such as the commutative property of addition, x+y=y+xx + y = y + x, and the associative property of multiplication, x(yz)=(xy)zx \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 G;1,1,\alg{G; 1, {}^{-1}, \cdot} consisting of:

  • a set GG, called the underlying set;
  • a distinguished element 1G1 \in G, called the identity element;
  • a unary operation 1:GG{}^{-1}: G \to G, written as xx1x \mapsto x^{-1}, called inversion;
  • a binary operation :G×GG\cdot : G \times G \to G, written as (x,y)xy(x, y) \mapsto x \cdot y, called the group operation or group product;

satisfying the following requirements:

  • Associative property: (xy)z=x(yz)(x \cdot y) \cdot z = x \cdot (y \cdot z) for all x,y,zGx, y, z \in G.
  • Identity property: 1x=x1=x1 \cdot x = x \cdot 1 = x for all xGx \in G.
  • Inverse property: xx1=x1x=1x \cdot x^{-1} = x^{-1} \cdot x = 1 for all xGx \in G.

Definition: Abelian Group

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

  • Commutative property: xy=yxx \cdot y = y \cdot x for all x,yGx, y \in G.