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Group (mathematics)

A group is an algebraic structure consisting of a set with a single binary operation that is associative, has an identity element, and where every element has an inverse. Key in symmetry and algebra.

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A group in mathematics is a basic algebraic object made of a collection of elements together with a rule for combining any two elements to get a third. The notion is intentionally abstract: it captures the essential features of many familiar systems, from the integers under addition to symmetries of geometric figures. For broader context see group theory and its connections to mathematics.

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Formal definition

Formally, a group is a set G equipped with a binary operation • (often written by juxtaposition) satisfying three axioms. These axioms are usually stated as:

  • Associativity: (ab)c = a(bc) for all a,b,c in G.
  • Identity: there exists e in G with ea = ae = a for every a in G.
  • Inverses: for each a in G there exists b in G with ab = ba = e.

The requirement that combining any two elements yields another element of the same set is called closure and is usually understood as part of the operation's definition.

Basic examples

Concrete instances help fix the idea. Typical examples include:

  • The integers with addition, where the identity is 0 and inverses are negation.
  • Nonzero real numbers with multiplication (identity 1, inverse 1/x).
  • Permutations of a finite set, composed by performing one permutation after another (symmetric groups).
  • Invertible matrices under matrix multiplication, important in linear algebra and geometry.

Important concepts and distinctions

Groups are classified and studied by many features. A group is called abelian if its operation is commutative (ab = ba for all elements); otherwise it is non-abelian. Subsets that themselves form groups are subgroups. Maps between groups that respect the operation are homomorphisms; kernels and images of homomorphisms play crucial roles. When a subgroup is normal, one can form a quotient group, an object that encapsulates how the subgroup sits inside the whole.

Historical notes and development

The idea of a group emerged in the 19th century from work on polynomial equations and permutation of roots. Mathematicians such as Évariste Galois and Arthur Cayley were central to early development: Galois used permutation structures to study solvability of equations, while later writers abstracted and systematized the axioms that define groups today. Over time the concept has become a unifying language across algebra, geometry and analysis.

Uses and significance

Groups express symmetry in physical systems and mathematical objects, underpin parts of number theory, topology and algebraic geometry, and appear in applications ranging from crystallography to particle physics and cryptography. Understanding the structure of groups—finite or infinite, simple or composite—guides classification problems and the study of invariants. For introductions and advanced treatments see algebraic structure resources and surveys in number theory and algebra.

For accessible expositions and further reading on specific families (permutation groups, matrix groups, Lie groups) consult specialized sources, or follow introductory expositions at set and structure summaries and online lecture notes about operations.

The turns of a Rubik's Cube form a group.
The turns of a Rubik's Cube form a group.

Introductory example

One of the best known groups is formed by the set of integers , \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}commonly denoted by , \mathbb {Z} together with addition.

The set of integers together with addition satisfies some basic properties:

  • For two integers aand the sum bis a+bagain an integer. On the other hand, if you were to divide two integers together, the result would mostly be a rational number and no longer an integer. Since this cannot happen with addition, the integers are said to be completed under addition.
  • For all integers a, band the associative law appliesc

(a + b) + c = a + (b + c).

In words, this means that it doesn't matter whether you first cadd aand bor band , the result is the same. This property is called associativity.

  • For any integer following appliesa

0 + a = a + 0 = a.

Therefore, the addition with zero does not change the initial number. Therefore, zero is called the neutral element of addition.

  • For every integer athere exists an integer bsuch that a + b = b + a = 0. That is, for every integer a there exists an integer bsuch that their sum is zero. In this case, the number bis called the inverse element of aand is -anotated as

These four properties of the set of integers together with their addition are generalized to other sets with a suitable operation in the definition of the group.

Examples

In the following some examples of groups are given. Thus groups of numbers, a group with exactly one element and examples of cyclic groups are given. Further examples of groups can be found in the list of small (finite) groups.

sets of numbers

  • The set of integers together with addition forms an (abelian) group. Together with multiplication, however, the set of integers is not a group (the inverse element of 2 would be 1/2).
  • The set of rational numbers \mathbb {Q} respectively the set of real numbers \mathbb {R} with addition is a group. Together with multiplication, the sets \mathbb{Q} \setminus \{0\}and \R \setminus \{0\}also groups.

The trivial group

→ Main article: Trivial group

The set which has only one element \{e\} can be considered as a group. Since every group has one neutral element, this one element must then be taken to be the neutral element. So then e * e = e. By means of this equality, the remaining group axioms can also be proved. The group with exactly one element is called the trivial group.

Cyclic groups

→ Main article: Cyclic group

A cyclic group is a group whose elements can be represented as powers of one of its elements. Using multiplicative notation, the elements of a cyclic group are

\ldots , a^{-3}, a^{-2}, a^{-1}, e = a^{0}, a, a^2, a^3, \ldots ,

where a^{-2} = a^{-1} \cdot a^{-1} and e denotes the neutral element of the group. The element ais called the producer or primitive root of the group. In additive notation, an element is a primitive root if the elements of the group are represented by

\ldots, -a -a, -a, 0, a, a+a, \ldots

can be displayed.

For example, the additive group of integers considered in the first section is a cyclic group with primitive root 1 . This group has infinitely many elements. In contrast, the multiplicative group of nth complex unit roots has finitely many elements. This group consists of all complex numbers zthat satisfy the equation

z^{n}=1

satisfy. The ngroup elements can be visualized as vertices of a regularn-corner. For n = 6 this is done in the graph on the right. The group operation is the multiplication of the complex numbers. So in the right-hand picture, multiplying by corresponds to zrotating the polygon counterclockwise by 60^{\circ }.

Cyclic groups have the property of being uniquely determined by the number of their elements. That is, two cyclic groups each with n elements are isomorphic, so a group isomorphism can be found between these two groups. So, in particular, all cyclic groups with infinitely many elements are equivalent to the cyclic group (\Z,+)of integers.

Symmetrical groups

→ Main article: Symmetric group

The symmetric group {\displaystyle S_{n}} consists of all permutations (permutations) of an n-elementary set. The group operation is the composition \circ (successive execution) of the permutations, the neutral element is the identical mapping. The symmetric group {\displaystyle S_{n}}is finite and has order {\displaystyle n!}. It is not abelian for . n\geq 3

The 6th complex unit roots can be taken as a cyclic group.
The 6th complex unit roots can be taken as a cyclic group.

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