Group — Mathematics Knowledge System

Overview

A group is a set equipped with a single operation that combines two elements to produce a third, satisfying four properties: closure, associativity, an identity element, and inverses. Groups are the simplest and most fundamental algebraic structure — the integers under addition, non-zero rationals under multiplication, and the symmetries of a shape are all groups, despite looking completely different on the surface.

Intuition

Think about the symmetries of a square: you can rotate it 90°, 180°, 270°, or flip it — 8 total operations that leave the square looking the same. Combine any two of these symmetries (do one, then another) and you always land on another symmetry from the same list of 8 — never something outside it. That closure, plus the fact that 'do nothing' is always an option and every move can be undone, is exactly what makes a set of symmetries a group. Groups are the mathematics of 'structured, undoable, combinable actions'.

Formal Definition

Definition

A group (G, ·) is a set G with a binary operation · satisfying four axioms:

\forall a,b \in G,\ a \cdot b \in G

Closure

(a \cdot b) \cdot c = a \cdot (b \cdot c)

Associativity

\exists e \in G:\ e \cdot a = a \cdot e = a\ \forall a

Identity

\forall a \in G,\ \exists a^{-1} \in G:\ a \cdot a^{-1} = e

Inverses

Notation

Notation	Meaning
$(G, \cdot)$	A group: set G with operation ·
$e$	The identity element
$a^{-1}$	The inverse of element a
$\|G\|$	Order of the group — number of elements (if finite)

Properties

Abelian (commutative) group

\text{A group where } a \cdot b = b \cdot a \text{ for all } a, b.

Example: (ℤ, +) is abelian; symmetries of a square generally are not.

Uniqueness of identity and inverses

\text{The identity element and each element's inverse are unique.}

Subgroup

\text{A subset } H \subseteq G \text{ that is itself a group under the same operation.}

Theorems

Theorem 1: Lagrange's Theorem

\text{For a finite group } G \text{ and subgroup } H, |H| \text{ divides } |G|.

Applications

Cryptographic algorithms (RSA, elliptic curve cryptography) rely on the group structure of modular arithmetic and elliptic curves.

Worked Examples

Closure: sum of two integers is an integer.
$a + b \in \mathbb{Z}$
Associativity holds for ordinary addition.
$(a+b)+c = a+(b+c)$
Identity: 0, since a + 0 = a.
$e = 0$
Inverse of a is -a, since a + (-a) = 0.
$a^{-1} = -a$

Answer: Yes — all four group axioms hold, and it's abelian since addition commutes.

Practice Problems

Difficulty 5/10

Which of the following is NOT a group under the given operation?

(ℤ, +)(ℚ \ {0}, ×)(ℕ, +)({1, -1}, ×)

Difficulty 5/10

The rotations of a square by 0°, 90°, 180°, 270° form a group under composition. How many elements does it have, and what is the inverse of the 90° rotation?

Difficulty 6/10

Diffie–Hellman key exchange works in the group of nonzero integers mod a prime p under multiplication. Why must this set form a GROUP for the scheme to work?

Common Mistakes

Common Mistake

Assuming every group is commutative (abelian).

Commutativity is NOT one of the four group axioms. Symmetry groups of shapes and matrix multiplication groups are common non-abelian examples.

Quiz

Which of these is NOT one of the four group axioms?

ClosureCommutativityAssociativityExistence of inverses

Flashcards

1 / 2

Historical Background

The concept crystallized from Évariste Galois's work in the 1830s on which polynomial equations can be solved by radicals — he used permutation groups to answer a 350-year-old open question. Arthur Cayley gave the first abstract definition of a group independent of permutations in 1854. The theory exploded in the late 19th and 20th centuries, becoming the backbone of modern abstract algebra and eventually physics (particle symmetries) and cryptography.

1832
Galois develops group-theoretic ideas the night before his fatal duel, in a letter to a friend
Évariste Galois
1854
Cayley gives the first abstract axiomatic definition of a group
Arthur Cayley

Summary

A group (G, ·) satisfies closure, associativity, has an identity, and every element has an inverse.
Commutativity is NOT required — groups where it holds are called abelian.
Symmetries of shapes, integers under addition, and modular arithmetic are all groups.
Lagrange's theorem: subgroup order always divides group order (for finite groups).
Groups underlie cryptography (RSA, elliptic curves) and particle physics symmetries.

References

PaperCayley, A. (1854). On the theory of groups, as depending on the symbolic equation θⁿ=1.
WebsiteWikipedia — Group (mathematics)