S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 2: Group Theory Basics

2.1 The group axioms

Groups are the central object of abstract algebra, capturing the essence of symmetry and invertible operations. The definition is deceptively simple—three axioms—yet groups appear everywhere: in geometry (symmetries), number theory (modular arithmetic), topology (fundamental groups), physics (gauge theories), and even chemistry (molecular symmetries). This lecture develops the foundations rigorously.

Definition 2.1.1 (Groups). A group is a set together with a map called a law of composition (also binary operation). Writing for , the law of composition must satisfy three axioms:

  1. Associativity: .
  2. Identity: such that for all .
  3. Inverses: such that .

We write or simply for the group, and call the order of .

The axioms have immediate but important consequences that we now prove carefully.

Proposition 2.1.1 (Uniqueness of identity). The identity element is unique.

Proof (Uniqueness of identity). Suppose and are both identities. Then (since is an identity) and (since is an identity). Thus . ∎

Proposition 2.1.2 (Uniqueness of inverses). For each , the inverse is unique. We denote it (or in additive notation).

Proof (Uniqueness of inverses). Suppose and are both inverses of , so and . Then: ∎

Proposition 2.1.3 (Cancellation laws). In a group :

  1. Left cancellation: .
  2. Right cancellation: .

Proof of cancellation. For left cancellation: if , multiply both sides on the left by : , so , giving , i.e., . Right cancellation is similar. ∎

Proposition 2.1.4 (Inverse of products). For all : . Note the reversal of order!

Proof of inverse formula. We verify : . Similarly for the other side. ∎

Proposition 2.1.5 (Inverse of inverse). For all : .

Proof of double inverse. By definition, satisfies . This says is an inverse of , so by uniqueness, . ∎

Note. It is customary to use additive notation (, , ) for abelian groups and multiplicative notation (, or , ) in general. This does not mean that multiplication implies non-commutativity!

Groups sit in a hierarchy of algebraic structures with fewer or more axioms.

Definition 2.1.2 (Abelian groups). A group is abelian (or commutative) if for all .

Definition 2.1.3 (Monoid). A monoid is a set with an associative binary operation and an identity element, but inverses are not required. Example: with identity 0, or with identity 1.

Definition 2.1.4 (Semigroup). A semigroup is a set with an associative binary operation—no identity or inverses required. Example: .

Note. There’s a pattern here: Group = Monoid + Inverses = Semigroup + Identity + Inverses. Category theory provides a unifying framework: a group is a category with one object where every morphism is invertible; a monoid is a category with one object (morphisms need not be invertible).