Group Definition

1Introduction

A group is one of the most fundamental structures in mathematics. It provides a formal way to describe symmetry, transformations, and operations that can be combined and reversed. Groups appear throughout mathematics, from number theory and geometry to modern physics and cryptography.

At its core, a group consists of a set together with an operation that combines any two elements of that set to produce another element of the same set. The operation must satisfy a small collection of rules, known as the group axioms, which ensure that the structure behaves in a predictable and useful way.

Formally, a group consists of a set of elements $G$ together with a binary operation $*$ defined on that set. We denote this structure by

$$\langle G, * \rangle$$

The operation $*$ must satisfy the following axioms.

2Closure

The group is closed under the operation. Combining any two elements of the group produces another element of the group.

$$\forall{x}\forall{y}[{{x}\in{G}}\land{{y}\in{G}} \implies (x*y)\in{G}]$$

3Associativity

The operation is associative for all elements of the group.

$$\forall{x}\forall{y}\forall{z}[x\in{G} \land y\in{G} \land z\in{G} \implies (x*y)*z = x*(y*z)]$$

4Identity Element

There exists an identity element $e$ such that combining it with any element leaves that element unchanged.

$$\exists{e}[e\in{G} \land \forall{x}(x\in{G} \implies e * x = x * e = x)]$$

5Inverse Elements

For every element in the group, there exists an inverse element whose combination with the original element yields the identity.

$$\forall{x}[x\in{G} \implies \exists{x^{-1}}(x^{-1}\in{G} \land x*x^{-1}=x^{-1}*x = e)]$$

A set equipped with an operation satisfying these four axioms is called a group.