Sets, Relations and Groups

set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.[5] The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.

Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.

History

The concept of a set emerged in mathematics at the end of the 19th century.[7] The German word for set, Menge, was coined by Bernard Bolzano in his work Paradoxes of the Infinite.

Passage with a translation of the original set definition of Georg Cantor. The German word Menge for set is translated with aggregate here.

Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:[1]

A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

Bertrand Russell called a set a class:[11]

When mathematicians deal with what they call a manifold, aggregate, Mengeensemble, or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case is the class.

Naive set theory

The foremost property of a set is that it can have elements, also called members. Two sets are equal when they have the same elements. More precisely, sets A and B are equal if every element of A is an element of B, and every element of B is an element of A; this property is called the extensionality of sets.[12]

The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:

• Russell’s paradox shows that the “set of all sets that do not contain themselves“, i.e., {x | x is a set and x ∉ x}, cannot exist.
• Cantor’s paradox shows that “the set of all sets” cannot exist.

Naïve set theory defines a set as any well-defined collection of distinct elements, but problems arise from the vagueness of the term well-defined.

Axiomatic set theory

In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axiomsAxiomatic set theory takes the concept of a set as a primitive notion.[13] The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to Gödel’s incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.[citation needed]

How sets are defined and set notation

Mathematical texts commonly denote sets by capital letters[14][5] in italic, such as ABC.[15] A set may also be called a collection or family, especially when its elements are themselves sets.

Roster notation

Roster or enumeration notation defines a set by listing its elements between curly brackets, separated by commas:[16][17][18][19]

A = {4, 2, 1, 3}
B = {blue, white, red}.

In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, {2, 4, 6} and {4, 6, 4, 2} represent the same set.[20][15][21]

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis ‘‘.[22][23] For instance, the set of the first thousand positive integers may be specified in roster notation as

{1, 2, 3, …, 1000}.

Infinite sets in roster notation

An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is

{0, 1, 2, 3, 4, …},

and the set of all integers is

{…, −3, −2, −1, 0, 1, 2, 3, …}.

Semantic definition

Another way to define a set is to use a rule to determine what the elements are:

Let A be the set whose members are the first four positive integers.
Let B be the set of colors of the French flag.

Such a definition is called a semantic description.[24][25]

Set-builder notation

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements.[25][26][27] For example, a set F can be defined as follows:

{\displaystyle F=\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.}

In this notation, the vertical bar “|” means “such that”, and the description can be interpreted as “F is the set of all numbers n such that n is an integer in the range from 0 to 19 inclusive”. Some authors use a colon “:” instead of the vertical bar.[28]

Classifying methods of definition

Philosophy uses specific terms to classify types of definitions:

• An intensional definition uses a rule to determine membership. Semantic definitions and definitions using set-builder notation are examples.
• An extensional definition describes a set by listing all its elements.[25] Such definitions are also called enumerative.
• An ostensive definition is one that describes a set by giving examples of elements; a roster involving an ellipsis would be an example.

Membership

If B is a set and x is an element of B, this is written in shorthand as x ∈ B, which can also be read as “x belongs to B“, or “x is in B“.[12] The statement “y is not an element of B” is written as y ∉ B, which can also be read as “y is not in B“.[29][30]

For example, with respect to the sets A = {1, 2, 3, 4}B = {blue, white, red}, and F = {n | n is an integer, and 0 ≤ n ≤ 19},

4 ∈ A and 12 ∈ F; and
20 ∉ F and green ∉ B.

The empty set

The empty set (or null set) is the unique set that has no members. It is denoted  or {\displaystyle \emptyset } or { }[31][32] or ϕ[33] (or ϕ).[34]

Singleton sets

singleton set is a set with exactly one element; such a set may also be called a unit set.[6] Any such set can be written as {x}, where x is the element. The set {x} and the element x mean different things; Halmos[35] draws the analogy that a box containing a hat is not the same as the hat.

Subsets

If every element of set A is also in B, then A is described as being a subset of B, or contained in B, written A ⊆ B,[36] or B ⊇ A.[37] The latter notation may be read B contains AB includes A, or B is a superset of A. The relationship between sets established by ⊆ is called inclusion or containment. Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B.[26]

If A is a subset of B, but A is not equal to B, then A is called a proper subset of B. This can be written A ⊊ B. Likewise, B ⊋ A means B is a proper superset of A, i.e. B contains A, and is not equal to A.

A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use A ⊂ B and B ⊃ A to mean A is any subset of B (and not necessarily a proper subset),[38][29] while others reserve A ⊂ B and B ⊃ A for cases where A is a proper subset of B.[36]

Examples:

• The set of all humans is a proper subset of the set of all mammals.
• {1, 3} ⊂ {1, 2, 3, 4}.
• {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

The empty set is a subset of every set,[31] and every set is a subset of itself:[38]

• ∅ ⊆ A.
• A ⊆ A.

Euler and Venn diagrams

A is a subset of B.
B is a superset of A.

An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If A is a subset of B, then the region representing A is completely inside the region representing B. If two sets have no elements in common, the regions do not overlap.

Venn diagram, in contrast, is a graphical representation of n sets in which the n loops divide the plane into 2n zones such that for each way of selecting some of the n sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are AB, and C, there should be a zone for the elements that are inside A and C and outside B (even if such elements do not exist).

Special sets of numbers in mathematics

The natural numbers  {\displaystyle \mathbb {N} } are contained in the integers {\displaystyle \mathbb {Z} }, which are contained in the rational numbers {\displaystyle \mathbb {Q} }, which are contained in the real numbers {\displaystyle \mathbb {R} }, which are contained in the complex numbers {\displaystyle \mathbb {C} }

There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.

Many of these important sets are represented in mathematical texts using bold (e.g. {\displaystyle {\mathbf {Z}}}) or blackboard bold (e.g. {\displaystyle \mathbb {Z} }) typeface.[39] These include

• {\displaystyle {\mathbf {N}}} or {\displaystyle \mathbb {N} }, the set of all natural numbers{\displaystyle {\mathbf {N}}=\{0,1,2,3,…\}} (often, authors exclude 0);[39]
• {\displaystyle {\mathbf {Z}}} or {\displaystyle \mathbb {Z} }, the set of all integers (whether positive, negative or zero): {\displaystyle {\mathbf {Z}}=\{…,-2,-1,0,1,2,3,…\}};[39]
• {\displaystyle {\mathbf {Q}}} or {\displaystyle \mathbb {Q} }, the set of all rational numbers (that is, the set of all proper and improper fractions): {\displaystyle {\mathbf {Q}}=\left\{{\frac {a}{b}}\mid a,b\in {\mathbf {Z}},b\neq 0\right\}}. For example, 7/4 ∈ Q and 5 = 5/1 ∈ Q;[39]
• {\displaystyle {\mathbf {R}}} or {\displaystyle \mathbb {R} }, the set of all real numbers, including all rational numbers and all irrational numbers (which include algebraic numbers such as {\displaystyle {\sqrt {2}}} that cannot be rewritten as fractions, as well as transcendental numbers such as π and e);[39]
• {\displaystyle {\mathbf {C}}} or {\displaystyle \mathbb {C} }, the set of all complex numbersC = {a + bi | ab ∈ R}, for example, 1 + 2i ∈ C.[39]

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, {\displaystyle \mathbf {Q} ^{+}} represents the set of positive rational numbers.

Functions

function (or mapping) from a set A to a set B is a rule that assigns to each “input” element of A an “output” that is an element of B; more formally, a function is a special kind of relation, one that relates each element of A to exactly one element of B. A function is called

• injective (or one-to-one) if it maps any two different elements of A to different elements of B,
• surjective (or onto) if for every element of B, there is at least one element of A that maps to it, and
• bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of A is paired with a unique element of B, and each element of B is paired with a unique element of A, so that there are no unpaired elements.

An injective function is called an injection, a surjective function is called a surjection, and a bijective function is called a bijection or one-to-one correspondence.

Cardinality

The cardinality of a set S, denoted |S|, is the number of members of S.[40] For example, if B = {blue, white, red}, then |B| = 3. Repeated members in roster notation are not counted,[41][42] so |{blue, white, red, blue, white}| = 3, too.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

The cardinality of the empty set is zero.[43]

Infinite sets and infinite cardinality

The list of elements of some sets is endless, or infinite. For example, the set {\displaystyle \mathbb {N} } of natural numbers is infinite.[26] In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have infinite cardinality.

Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers.[44] Sets with cardinality less than or equal to that of {\displaystyle \mathbb {N} } are called countable sets; these are either finite sets or countably infinite sets (sets of the same cardinality as {\displaystyle \mathbb {N} }); some authors use “countable” to mean “countably infinite”. Sets with cardinality strictly greater than that of {\displaystyle \mathbb {N} } are called uncountable sets.

However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.[45]

The continuum hypothesis

The continuum hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line.[46] In 1963, Paul Cohen proved that the continuum hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice.[47] (ZFC is the most widely-studied version of axiomatic set theory.)

Power sets

The power set of a set S is the set of all subsets of S.[26] The empty set and S itself are elements of the power set of S, because these are both subsets of S. For example, the power set of {1, 2, 3} is {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. The power set of a set S is commonly written as P(S) or 2S.[26][48][15]

If S has n elements, then P(S) has 2n elements.[49] For example, {1, 2, 3} has three elements, and its power set has 23 = 8 elements, as shown above.

If S is infinite (whether countable or uncountable), then P(S) is uncountable. Moreover, the power set is always strictly “bigger” than the original set, in the sense that any attempt to pair up the elements of S with the elements of P(S) will leave some elements of P(S) unpaired. (There is never a bijection from S onto P(S).)[50]

Partitions

partition of a set S is a set of nonempty subsets of S, such that every element x in S is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is S.[51][52]

Basic operation

There are several fundamental operations for constructing new sets from given sets.

Unions

The union of A and B, denoted A ∪ B

Two sets can be joined: the union of A and B, denoted by A ∪ B, is the set of all things that are members of A or of B or of both.

Examples:

• {1, 2} ∪ {1, 2} = {1, 2}.
• {1, 2} ∪ {2, 3} = {1, 2, 3}.
• {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.

Some basic properties of unions:

• A ∪ B = B ∪ A.
• A ∪ (B ∪ C) = (A ∪ B) ∪ C.
• A ⊆ (A ∪ B).
• A ∪ A = A.
• A ∪ ∅ = A.
• A ⊆ B if and only if A ∪ B = B.

Intersections

A new set can also be constructed by determining which members two sets have “in common”. The intersection of A and B, denoted by A ∩ B, is the set of all things that are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.

The intersection of A and B, denoted A ∩ B.

Examples:

• {1, 2} ∩ {1, 2} = {1, 2}.
• {1, 2} ∩ {2, 3} = {2}.
• {1, 2} ∩ {3, 4} = ∅.

Some basic properties of intersections:

• A ∩ B = B ∩ A.
• A ∩ (B ∩ C) = (A ∩ B) ∩ C.
• A ∩ B ⊆ A.
• A ∩ A = A.
• A ∩ ∅ = ∅.
• A ⊆ B if and only if A ∩ B = A.

Complements

The relative complement
of B in A

The complement of A in U

The symmetric difference of A and B

Two sets can also be “subtracted”. The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or A − B), is the set of all elements that are members of A, but not members of B. It is valid to “subtract” members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so will not affect the elements in the set.

In certain settings, all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′ or Ac.

• A′ = U \ A

Examples:

• {1, 2} \ {1, 2} = ∅.
• {1, 2, 3, 4} \ {1, 3} = {2, 4}.
• If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E = E′ = O.

Some basic properties of complements include the following:

• A \ B ≠ B \ A for A ≠ B.
• A ∪ A′ = U.
• A ∩ A′ = ∅.
• (A′)′ = A.
• ∅ \ A = ∅.
• A \ ∅ = A.
• A \ A = ∅.
• A \ U = ∅.
• A \ A′ = A and A′ \ A = A′.
• U′ = ∅ and ∅′ = U.
• A \ B = A ∩ B.
• if A ⊆ B then A \ B = ∅.

An extension of the complement is the symmetric difference, defined for sets AB as

{\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A).}

For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

Cartesian product

A new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (ab) such that a is a member of A and b is a member of B.

Examples:

• {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
• {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
• {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:

• A × ∅ = ∅.
• A × (B ∪ C) = (A × B) ∪ (A × C).
• (A ∪ B) × C = (A × C) ∪ (B × C).

Let A and B be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:

• |A × B| = |B × A| = |A| × |B|.

Applications

Sets are ubiquitous in modern mathematics. For example, structures in abstract algebra, such as groupsfields and rings, are sets closed under one or more operations.

One of the main applications of naive set theory is in the construction of relations. A relation from a domain A to a codomain B is a subset of the Cartesian product A × B. For example, considering the set S = {rock, paper, scissors} of shapes in the game of the same name, the relation “beats” from S to S is the set B = {(scissors,paper), (paper,rock), (rock,scissors)}; thus x beats y in the game if the pair (x,y) is a member of B. Another example is the set F of all pairs (xx2), where x is real. This relation is a subset of R × R, because the set of all squares is subset of the set of all real numbers. Since for every x in R, one and only one pair (x,…) is found in F, it is called a function. In functional notation, this relation can be written as F(x) = x2.

Principle of inclusion and exclusion

The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as

{\displaystyle |A\cup B|=|A|+|B|-|A\cap B|.}

A more general form of the principle can be used to find the cardinality of any finite union of sets:

{\displaystyle {\begin{aligned}\left|A_{1}\cup A_{2}\cup A_{3}\cup \ldots \cup A_{n}\right|=&\left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots \left|A_{n}\right|\right)\\&{}-\left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots \left|A_{n-1}\cap A_{n}\right|\right)\\&{}+\ldots \\&{}+\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap \ldots \cap A_{n}\right|\right).\end{aligned}}}

De Morgan’s laws

Augustus De Morgan stated two laws about sets.

If A and B are any two sets then,

• (A ∪ B)′ = A′ ∩ B

The complement of A union B equals the complement of A intersected with the complement of B.

• (A ∩ B)′ = A′ ∪ B

The complement of A intersected with B is equal to the complement of A union to the complement of B.

In mathematics, a binary relation is a general concept that defines some relation between the elements of two sets. It is a generalization of the more commonly understood idea of a mathematical function, but with fewer restrictions. A binary relation over sets X and Y is a set of ordered pairs (xy) consisting of elements x in X and y in Y.[1] It encodes the common concept of relation: an element x is related to an element yif and only if the pair (xy) belongs to the set of ordered pairs that defines the binary relation. A binary relation is the most studied special case n = 2 of an n-ary relation over sets X1, …, Xn, which is a subset of the Cartesian product X1 × … × Xn.[1]

A trivial example of a binary relation over set X of all real numbers ({\displaystyle \mathbb {R} }) and set Y of all real numbers ({\displaystyle \mathbb {R} }) is the set of all pairs for which elements {\displaystyle x=y}. This is equivalent to the function {\displaystyle y=f(x)=x}.

Another example of a binary relation is the “divides” relation over the set of prime numbers {\displaystyle \mathbb {P} } and the set of integers {\displaystyle \mathbb {Z} }, in which each prime p is related to each integer z that is a multiple of p, but not to an integer that is not a multiple of p. In this relation, for instance, the prime number 2 is related to numbers such as −4, 0, 6, 10, but not to 1 or 9, just as the prime number 3 is related to 0, 6, and 9, but not to 4 or 13.

Binary relations are used in many branches of mathematics to model a wide variety of concepts. These include, among others:

function may be defined as a special kind of binary relation.[2] Binary relations are also heavily used in computer science, such as in a relational database management system (RDBMS).

A binary relation over sets X and Y is an element of the power set of X × Y. Since the latter set is ordered by inclusion (⊆), each relation has a place in the lattice of subsets of X × Y. A binary relation is either a homogeneous relation or a heterogeneous relation depending on whether X = Y or not.

Since relations are sets, they can be manipulated using set operations, including unionintersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder,[3] Clarence Lewis,[4] and Gunther Schmidt.[5] A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice.

In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of “is an element of” or “is a subset of” in set theory, without running into logical inconsistencies such as Russell’s paradox.

The terms correspondence,[6] dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term “binary relation” for any subset of a Cartesian product X × Y without reference to X and Y, and reserve the term “correspondence” for a binary relation with reference to X and Y.

Definition

Given sets X and Y, the Cartesian product X × Y is defined as {(xy) | x ∈ X and y ∈ Y}, and its elements are called ordered pairs.

binary relation R over sets X and Y is a subset of X × Y.[1][7] The set X is called the domain[1] or set of departure of R, and the set Y the codomain or set of destination of R. In order to specify the choices of the sets X and Y, some authors define a binary relation or correspondence as an ordered triple (XYG), where G is a subset of X × Y called the graph of the binary relation. The statement (xy) ∈ R reads “x is R-related to y” and is written in infix notation as xRy.[3][4][5][note 1] The domain of definition or active domain[1] of R is the set of all x such that xRy for at least one y. The codomain of definitionactive codomain,[1] image or range of R is the set of all y such that xRy for at least one x. The field of R is the union of its domain of definition and its codomain of definition.[9][10][11]

When X = Y, a binary relation is called a homogeneous relation (or endorelation). Otherwise it is a heterogeneous relation.[12][13][14]

In a binary relation, the order of the elements is important; if x ≠ y then yRx can be true or false independently of xRy. For example, 3 divides 9, but 9 does not divide 3.

Example

2nd example relation
A
B
ballcardollcup
John+
Mary+
Venus+
1st example relation
A
B
ballcardollcup
John+
Mary+
Ian
Venus+

The following example shows that the choice of codomain is important. Suppose there are four objects A = {ball, car, doll, cup} and four people B = {John, Mary, Ian, Venus}. A possible relation on A and B is the relation “is owned by”, given by R = {(ball, John), (doll, Mary), (car, Venus)}. That is, John owns the ball, Mary owns the doll, and Venus owns the car. Nobody owns the cup and Ian owns nothing, see 1st example. As a set, R does not involve Ian, and therefore R could have been viewed as a subset of A × {John, Mary, Venus}, i.e. a relation over A and {John, Mary, Venus}, see 2nd example. While the 2nd example relation is surjective (see below), the 1st is not.

Special types of binary relations

Examples of four types of binary relations over the real numbers: one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black).

Some important types of binary relations R over sets X and Y are listed below.

Uniqueness properties:

Injective (also called left-unique)[15]
For all xz ∈ X and all y ∈ Y, if xRy and zRy then x = z. For such a relation, {Y} is called primary key of R.[1] For example, the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both −1 and 1 to 1), nor the black one (as it relates both −1 and 1 to 0).
Functional (also called right-unique,[15]right-definite[16] or univalent)
[5] For all x ∈ X and all yz ∈ Y, if xRy and xRz then y = z. Such a binary relation is called a partial function. For such a relation, {X} is called a primary key of R.[1] For example, the red and green binary relations in the diagram are functional, but the blue one is not (as it relates 1 to both −1 and 1), nor the black one (as it relates 0 to both −1 and 1).
One-to-one
Injective and functional. For example, the green binary relation in the diagram is one-to-one, but the red, blue and black ones are not.
One-to-many
Injective and not functional. For example, the blue binary relation in the diagram is one-to-many, but the red, green and black ones are not.
Many-to-one
Functional and not injective. For example, the red binary relation in the diagram is many-to-one, but the green, blue and black ones are not.
Many-to-many
Not injective nor functional. For example, the black binary relation in the diagram is many-to-many, but the red, green and blue ones are not.

Totality properties (only definable if the domain X and codomain Y are specified):

Total (also called left-total)
For all x in X there exists a y in Y such that xRy. In other words, the domain of definition of R is equal to X. This property is different from the definition of connected (also called total by some authors)[citation needed] in the section Properties. Such a binary relation is called a multivalued function. For example, the red and green binary relations in the diagram are total, but the blue one is not (as it does not relate −1 to any real number), nor the black one (as it does not relate 2 to any real number).
Surjective (also called right-total[15] or onto)
For all y in Y, there exists an x in X such that xRy. In other words, the codomain of definition of R is equal to Y. For example, the green and blue binary relations in the diagram are surjective, but the red one is not (as it does not relate any real number to −1), nor the black one (as it does not relate any real number to 2).

Uniqueness and totality properties (only definable if the domain X and codomain Y are specified):

function
A binary relation that is functional and total. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.
An injection
A function that is injective. For example, the green binary relation in the diagram is an injection, but the red, blue and black ones are not.
surjection
A function that is surjective. For example, the green binary relation in the diagram is a surjection, but the red, blue and black ones are not.
bijection
A function that is injective and surjective. For example, the green binary relation in the diagram is a bijection, but the red, blue and black ones are not.

Operations on binary relations

Union

If R and S are binary relations over sets X and Y then R ∪ S = {(xy) | xRy or xSy} is the union relation of R and S over X and Y.

The identity element is the empty relation. For example, ≤ is the union of < and =, and ≥ is the union of > and =.

Intersection

If R and S are binary relations over sets X and Y then R ∩ S = {(xy) | xRy and xSy} is the intersection relation of R and S over X and Y.

The identity element is the universal relation. For example, the relation “is divisible by 6” is the intersection of the relations “is divisible by 3” and “is divisible by 2”.

Composition

If R is a binary relation over sets X and Y, and S is a binary relation over sets Y and Z then S ∘ R = {(xz) | there exists y ∈ Y such that xRy and ySz} (also denoted by RS) is the composition relation of R and S over X and Z.

The identity element is the identity relation. The order of R and S in the notation S ∘ R, used here agrees with the standard notational order for composition of functions. For example, the composition “is mother of” ∘ “is parent of” yields “is maternal grandparent of”, while the composition “is parent of” ∘ “is mother of” yields “is grandmother of”. For the former case, if x is the parent of y and y is the mother of z, then x is the maternal grandparent of z.

Converse

If R is a binary relation over sets X and Y then RT = {(yx) | xRy} is the converse relation of R over Y and X.

For example, = is the converse of itself, as is ≠, and < and > are each other’s converse, as are ≤ and ≥. A binary relation is equal to its converse if and only if it is symmetric.

Complement

If R is a binary relation over sets X and Y then R = {(xy) | not xRy} (also denoted by R or ¬ R) is the complementary relation of R over X and Y.

For example, = and ≠ are each other’s complement, as are ⊆ and ⊈, ⊇ and ⊉, and ∈ and ∉, and, for total orders, also < and ≥, and > and ≤.

The complement of the converse relation RT is the converse of the complement: {\displaystyle {\overline {R^{\mathsf {T}}}}={\bar {R}}^{\mathsf {T}}.}

If X = Y, the complement has the following properties:

• If a relation is symmetric, then so is the complement.
• The complement of a reflexive relation is irreflexive—and vice versa.
• The complement of a strict weak order is a total preorder—and vice versa.

Restriction

If R is a binary homogeneous relation over a set X and S is a subset of X then R|S = {(xy) | xRy and x ∈ S and y ∈ S} is the restriction relation of R to S over X.

If R is a binary relation over sets X and Y and if S is a subset of X then R|S = {(xy) | xRy and x ∈ S} is the left-restriction relation of R to S over X and Y.

If R is a binary relation over sets X and Y and if S is a subset of Y then R|S = {(xy) | xRy and y ∈ S} is the right-restriction relation of R to S over X and Y.

If a relation is reflexive, irreflexive, symmetricantisymmetricasymmetrictransitivetotaltrichotomous, a partial ordertotal orderstrict weak ordertotal preorder (weak order), or an equivalence relation, then so too are its restrictions.

However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation “x is parent of y” to females yields the relation “x is mother of the woman y“; its transitive closure doesn’t relate a woman with her paternal grandmother. On the other hand, the transitive closure of “is parent of” is “is ancestor of”; its restriction to females does relate a woman with her paternal grandmother.

Also, the various concepts of completeness (not to be confused with being “total”) do not carry over to restrictions. For example, over the real numbers a property of the relation ≤ is that every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation ≤ to the rational numbers.

A binary relation R over sets X and Y is said to be contained in a relation S over X and Y, written {\displaystyle R\subseteq S,} if R is a subset of S, that is, for all {\displaystyle x\in X} and {\displaystyle y\in Y,} if xRy, then xSy. If R is contained in S and S is contained in R, then R and S are called equal written R = S. If R is contained in S but S is not contained in R, then R is said to be smaller than S, written R ⊊ S. For example, on the rational numbers, the relation > is smaller than ≥, and equal to the composition > ∘ >.

Homogeneous relation

homogeneous relation(also called endorelation) over a set X is a binary relation over X and itself, i.e. it is a subset of the Cartesian product X × X.[14][17][18] It is also simply called a (binary) relation over X. An example of a homogeneous relation is the relation of kinship, where the relation is over people.

A homogeneous relation R over a set X may be identified with a directed simple graph permitting loops, or if it is symmetric, with an undirected simple graph permitting loops, where X is the vertex set and R is the edge set (there is an edge from a vertex x to a vertex y if and only if xRy). It is called the adjacency relation of the graph.

The set of all homogeneous relations {\displaystyle {\mathcal {B}}(X)} over a set X is the set 2X × X which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on {\displaystyle {\mathcal {B}}(X)}, it forms a semigroup with involution.

Particular homogeneous relations

Some important particular homogeneous relations over a set X are:

The empty relation
E =  ⊆ X × X;
The universal relation
U = X × X;
The identity relation
I = {(xx) | x ∈ X}.

For arbitrary elements x and y of X:

• xEy holds never;
• xUy holds always;
• xIy holds if and only if x = y.

Properties

Some important properties that a homogeneous relation R over a set X may have are:

Reflexive
for all x ∈ XxRx. For example, ≥ is a reflexive relation but > is not.
Irreflexive (or strict)
for all x ∈ X, not xRx. For example, > is an irreflexive relation, but ≥ is not.

The previous 2 alternatives are not exhaustive; e.g., the red binary relation y = x2 given in the section § Special types of binary relations is neither irreflexive, nor reflexive, since it contains the pair (0, 0), but not (2, 2), respectively.

Symmetric
for all xy ∈ X, if xRy then yRx. For example, “is a blood relative of” is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x.
Antisymmetric
for all xy ∈ X, if xRy and yRx then x = y. For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false).[19]
Asymmetric
for all xy ∈ X, if xRy then not yRx. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.[20] For example, > is an asymmetric relation, but ≥ is not.

Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation xRy defined by x > 2 is neither symmetric nor antisymmetric, let alone asymmetric.

Transitive
for all xyz ∈ X, if xRy and yRz then xRz. A transitive relation is irreflexive if and only if it is asymmetric.[21] For example, “is ancestor of” is a transitive relation, while “is parent of” is not.
Dense
for all xy ∈ X such that xRy, there exists some z ∈ X such that xRz and zRy. This is used in dense orders.
Connected
for all xy ∈ X, if x ≠ y then xRy or yRx. This property is sometimes called “total”, which is distinct from the definitions of “total” given in the section Special types of binary relations.
Strongly connected
for all xy ∈ XxRy or yRx. This property is sometimes called “total”, which is distinct from the definitions of “total” given in the section Special types of binary relations.
Trichotomous
for all xy ∈ X, exactly one of xRyyRx or x = y holds. For example, > is a trichotomous relation, while the relation “divides” over the natural numbers is not.[22]
Serial (or left-total)
for all x ∈ X, there exists some y ∈ X such that xRy. For example, > is a serial relation over the integers. But it is not a serial relation over the positive integers, because there is no y in the positive integers such that 1 > y.[23] However, < is a serial relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is serial: for a given x, choose y = x.
Well-founded
every nonempty subset S of X contains a minimal element with respect to R. Well-foundedness implies the descending chain condition (that is, no infinite chain … xnRRx3Rx2Rx1 can exist). If the axiom of dependent choice is assumed, both conditions are equivalent.[24][25]
Preorder

A relation that is reflexive and transitive.

Total preorder (also, linear preorder or weak order)
A relation that is reflexive, transitive, and connected.
Partial order (also, order[citation needed])

A relation that is reflexive, antisymmetric, and transitive.

Strict partial order (also, strict order[citation needed])
A relation that is irreflexive, antisymmetric, and transitive.
Total order (also, linear ordersimple order, or chain)
A relation that is reflexive, antisymmetric, transitive and connected.[26]
Strict total order (also, strict linear orderstrict simple order, or strict chain)
A relation that is irreflexive, antisymmetric, transitive and connected.
Partial equivalence relation

A relation that is symmetric and transitive.

Equivalence relation
A relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and serial, since these properties imply reflexivity.

Operations

If R is a homogeneous relation over a set X then each of the following is a homogeneous relation over X:

Reflexive closure
R=, defined as R= = {(xx) | x ∈ X} ∪ R or the smallest reflexive relation over X containing R. This can be proven to be equal to the intersection of all reflexive relations containing R.
Reflexive reduction
R, defined as R = R \ {(xx) | x ∈ X} or the largest irreflexive relation over X contained in R.
Transitive closure
R+, defined as the smallest transitive relation over X containing R. This can be seen to be equal to the intersection of all transitive relations containing R.
Reflexive transitive closure
R*, defined as R* = (R+)=, the smallest preorder containing R.
Reflexive transitive symmetric closure
R, defined as the smallest equivalence relation over X containing R.

All operations defined in the section § Operations on binary relations also apply to homogeneous relations.

Homogeneous relations by property
ReflexivitySymmetryTransitivityConnectednessSymbolExample
Directed graph
Undirected graph Symmetric
DependencyReflexiveSymmetric
TournamentIrreflexiveAntisymmetric   Pecking order
PreorderReflexive Yes Preference
Total preorderReflexive YesYes
Partial orderReflexiveAntisymmetricYes Subset
Strict partial orderIrreflexiveAntisymmetricYes <Strict subset
Total orderReflexiveAntisymmetricYesYesAlphabetical order
Strict total orderIrreflexiveAntisymmetricYesYes<Strict alphabetical order
Partial equivalence relation SymmetricYes
Equivalence relationReflexiveSymmetricYes ∼, ≡Equality

Examples

In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it was elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.[1][2]

In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a general group. Lie groups appear in symmetry groups in geometry, and also in the Standard Model of particle physics. The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativityPoint groups describe symmetry in molecular chemistry.

The concept of a group arose in the study of polynomial equations, starting with Évariste Galois in the 1830s, who introduced the term group (French: groupe) for the symmetry group of the roots of an equation, now called a Galois group. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroupsquotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory.

Definition and illustration

First example: the integers

One of the more familiar groups is the set of integers

{\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}}

together with addition.[3] For any two integers {\displaystyle a} and {\displaystyle b}, the sum {\displaystyle a+b} is also an integer; this closure property says that {\displaystyle +} is a binary operation on {\displaystyle \mathbb {Z} }. The following properties of integer addition serve as a model for the group axioms in the definition below.

• For all integers {\displaystyle a}{\displaystyle b} and {\displaystyle c}, one has {\displaystyle (a+b)+c=a+(b+c)}. Expressed in words, adding {\displaystyle a} to {\displaystyle b} first, and then adding the result to {\displaystyle c} gives the same final result as adding {\displaystyle a} to the sum of {\displaystyle b} and {\displaystyle c}. This property is known as associativity.
• If {\displaystyle a} is any integer, then {\displaystyle 0+a=a} and {\displaystyle a+0=a}Zero is called the identity element of addition because adding it to any integer returns the same integer.
• For every integer {\displaystyle a}, there is an integer {\displaystyle b} such that {\displaystyle a+b=0} and {\displaystyle b+a=0}. The integer {\displaystyle b} is called the inverse element of the integer {\displaystyle a} and is denoted {\displaystyle -a}.

The integers, together with the operation {\displaystyle +}, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.

Definition

The axioms for a group are short and natural… Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.

Richard Borcherds in Mathematicians: An Outer View of the Inner World[4]

A group is a set {\displaystyle G} together with a binary operation on {\displaystyle G}, here denoted “{\displaystyle \cdot }“, that combines any two elements {\displaystyle a} and {\displaystyle b} to form an element of {\displaystyle G}, denoted {\displaystyle a\cdot b}, such that the following three requirements, known as group axioms, are satisfied:[5][6][7][a]

Associativity
For all {\displaystyle a}{\displaystyle b}{\displaystyle c} in {\displaystyle G}, one has {\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}.
Identity element
There exists an element {\displaystyle e} in {\displaystyle G} such that, for every {\displaystyle a} in {\displaystyle G}, one has {\displaystyle e\cdot a=a} and {\displaystyle a\cdot e=a}.
Such an element is unique (see below). It is called the identity element of the group.
Inverse element
For each {\displaystyle a} in {\displaystyle G}, there exists an element {\displaystyle b} in {\displaystyle G} such that {\displaystyle a\cdot b=e} and {\displaystyle b\cdot a=e}, where {\displaystyle e} is the identity element.
For each {\displaystyle a}, the element {\displaystyle b} is unique (see below); it is called the inverse of {\displaystyle a} and is commonly denoted {\displaystyle a^{-1}}.

Notation and terminology

Formally, the group is the ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the underlying set of the group, and the operation is called the group operation or the group law.

A group and its underlying set are thus two different mathematical objects. To avoid cumbersome notation, it is common to abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation.

For example, consider the set of real numbers {\displaystyle \mathbb {R} }, which has the operations of addition {\displaystyle a+b} and multiplication {\displaystyle ab}. Formally, {\displaystyle \mathbb {R} } is a set, {\displaystyle (\mathbb {R} ,+)} is a group, and {\displaystyle (\mathbb {R} ,+,\cdot )} is a field. But it is common to write {\displaystyle \mathbb {R} } to denote any of these three objects.

The additive group of the field {\displaystyle \mathbb {R} } is the group whose underlying set is {\displaystyle \mathbb {R} } and whose operation is addition. The multiplicative group of the field {\displaystyle \mathbb {R} } is the group {\displaystyle \mathbb {R} ^{\times }} whose underlying set is the set of nonzero real numbers {\displaystyle \mathbb {R} \smallsetminus \{0\}} and whose operation is multiplication.

More generally, one speaks of an additive group whenever the group operation is notated as addition; in this case, the identity is typically denoted {\displaystyle 0}, and the inverse of an element {\displaystyle x} is denoted {\displaystyle -x}. Similarly, one speaks of a multiplicative group whenever the group operation is notated as multiplication; in this case, the identity is typically denoted {\displaystyle 1}, and the inverse of an element {\displaystyle x} is denoted {\displaystyle x^{-1}}. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, {\displaystyle ab} instead of {\displaystyle a\cdot b}.

The definition of a group does not require that {\displaystyle a\cdot b=b\cdot a} for all elements {\displaystyle a} and {\displaystyle b} in {\displaystyle G}. If this additional condition holds, then the operation is said to be commutative, and the group is called an abelian group. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used.

Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are functions, the operation is often function composition {\displaystyle f\circ g}; then the identity may be denoted id. In the more specific cases of geometric transformation groups, symmetry groups, permutation groups, and automorphism groups, the symbol {\displaystyle \circ } is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.

Second example: a symmetry group

Two figures in the plane are congruent if one can be changed into the other using a combination of rotationsreflections, and translations. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetries. A square has eight symmetries. These are:

 {\displaystyle \mathrm {id} } (keeping it as it is) {\displaystyle r_{1}} (rotation by 90° clockwise) {\displaystyle r_{2}} (rotation by 180°) {\displaystyle r_{3}} (rotation by 270° clockwise) {\displaystyle f_{\mathrm {v} }} (vertical reflection) {\displaystyle f_{\mathrm {h} }} (horizontal reflection) {\displaystyle f_{\mathrm {d} }} (diagonal reflection) {\displaystyle f_{\mathrm {c} }} (counter-diagonal reflection)
• the identity operation leaving everything unchanged, denoted id;
• rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by {\displaystyle r_{1}}{\displaystyle r_{2}} and {\displaystyle r_{3}}, respectively;
• reflections about the horizontal and vertical middle line ({\displaystyle f_{\mathrm {v} }} and {\displaystyle f_{\mathrm {h} }}), or through the two diagonals ({\displaystyle f_{\mathrm {d} }} and {\displaystyle f_{\mathrm {c} }}).

These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example, {\displaystyle r_{1}} sends a point to its rotation 90° clockwise around the square’s center, and {\displaystyle f_{\mathrm {h} }} sends a point to its reflection across the square’s vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called the dihedral group of degree four, denoted {\displaystyle \mathrm {D} _{4}}. The underlying set of the group is the above set of symmetries, and the group operation is function composition.[8] Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first {\displaystyle a} and then {\displaystyle b} is written symbolically from right to left as {\displaystyle b\circ a} (“apply the symmetry {\displaystyle b} after performing the symmetry {\displaystyle a}“). This is the usual notation for composition of functions.

The group table lists the results of all such compositions possible. For example, rotating by 270° clockwise ({\displaystyle r_{3}}) and then reflecting horizontally ({\displaystyle f_{\mathrm {h} }}) is the same as performing a reflection along the diagonal ({\displaystyle f_{\mathrm {d} }}). Using the above symbols, highlighted in blue in the group table:

{\displaystyle f_{\mathrm {h} }\circ r_{3}=f_{\mathrm {d} }.}

Group table of {\displaystyle \mathrm {D} _{4}}
{\displaystyle \circ }{\displaystyle \mathrm {id} }{\displaystyle r_{1}}{\displaystyle r_{2}}{\displaystyle r_{3}}{\displaystyle f_{\mathrm {v} }}{\displaystyle f_{\mathrm {h} }}{\displaystyle f_{\mathrm {d} }}{\displaystyle f_{\mathrm {c} }}
{\displaystyle \mathrm {id} }{\displaystyle \mathrm {id} }{\displaystyle r_{1}}{\displaystyle r_{2}}{\displaystyle r_{3}}{\displaystyle f_{\mathrm {v} }}{\displaystyle f_{\mathrm {h} }}{\displaystyle f_{\mathrm {d} }}{\displaystyle f_{\mathrm {c} }}
{\displaystyle r_{1}}{\displaystyle r_{1}}{\displaystyle r_{2}}{\displaystyle r_{3}}{\displaystyle \mathrm {id} }{\displaystyle f_{\mathrm {c} }}{\displaystyle f_{\mathrm {d} }}{\displaystyle f_{\mathrm {v} }}{\displaystyle f_{\mathrm {h} }}
{\displaystyle r_{2}}{\displaystyle r_{2}}{\displaystyle r_{3}}{\displaystyle \mathrm {id} }{\displaystyle r_{1}}{\displaystyle f_{\mathrm {h} }}{\displaystyle f_{\mathrm {v} }}{\displaystyle f_{\mathrm {c} }}{\displaystyle f_{\mathrm {d} }}
{\displaystyle r_{3}}{\displaystyle r_{3}}{\displaystyle \mathrm {id} }{\displaystyle r_{1}}{\displaystyle r_{2}}{\displaystyle f_{\mathrm {d} }}{\displaystyle f_{\mathrm {c} }}{\displaystyle f_{\mathrm {h} }}{\displaystyle f_{\mathrm {v} }}
{\displaystyle f_{\mathrm {v} }}{\displaystyle f_{\mathrm {v} }}{\displaystyle f_{\mathrm {d} }}{\displaystyle f_{\mathrm {h} }}{\displaystyle f_{\mathrm {c} }}{\displaystyle \mathrm {id} }{\displaystyle r_{2}}{\displaystyle r_{1}}{\displaystyle r_{3}}
{\displaystyle f_{\mathrm {h} }}{\displaystyle f_{\mathrm {h} }}{\displaystyle f_{\mathrm {c} }}{\displaystyle f_{\mathrm {v} }}{\displaystyle f_{\mathrm {d} }}{\displaystyle r_{2}}{\displaystyle \mathrm {id} }{\displaystyle r_{3}}{\displaystyle r_{1}}
{\displaystyle f_{\mathrm {d} }}{\displaystyle f_{\mathrm {d} }}{\displaystyle f_{\mathrm {h} }}{\displaystyle f_{\mathrm {c} }}{\displaystyle f_{\mathrm {v} }}{\displaystyle r_{3}}{\displaystyle r_{1}}{\displaystyle \mathrm {id} }{\displaystyle r_{2}}
{\displaystyle f_{\mathrm {c} }}{\displaystyle f_{\mathrm {c} }}{\displaystyle f_{\mathrm {v} }}{\displaystyle f_{\mathrm {d} }}{\displaystyle f_{\mathrm {h} }}{\displaystyle r_{1}}{\displaystyle r_{3}}{\displaystyle r_{2}}{\displaystyle \mathrm {id} }
The elements {\displaystyle \mathrm {id} }{\displaystyle r_{1}}{\displaystyle r_{2}}, and {\displaystyle r_{3}} form a subgroup whose group table is highlighted in   red (upper left region). A left and right coset of this subgroup are highlighted in   green (in the last row) and   yellow (last column), respectively. The result of the composition {\displaystyle f_{\mathrm {h} }\circ r_{3}}, the symmetry {\displaystyle f_{\mathrm {d} }}, is highlighted in   blue (below table center).

Given this set of symmetries and the described operation, the group axioms can be understood as follows.

Binary operation: Composition is a binary operation. That is, {\displaystyle a\circ b} is a symmetry for any two symmetries {\displaystyle a} and {\displaystyle b}. For example,

{\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} },}

that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal ({\displaystyle f_{\mathrm {c} }}). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the group table.

Associativity: The associativity axiom deals with composing more than two symmetries: Starting with three elements {\displaystyle a}{\displaystyle b} and {\displaystyle c} of {\displaystyle \mathrm {D} _{4}}, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose {\displaystyle a} and {\displaystyle b} into a single symmetry, then to compose that symmetry with {\displaystyle c}. The other way is to first compose {\displaystyle b} and {\displaystyle c}, then to compose the resulting symmetry with {\displaystyle a}. These two ways must give always the same result, that is,

{\displaystyle (a\circ b)\circ c=a\circ (b\circ c),}

For example, {\displaystyle (f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}=f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})} can be checked using the group table:

{\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}}

Identity element: The identity element is {\displaystyle \mathrm {id} }, as it does not change any symmetry {\displaystyle a} when composed with it either on the left or on the right.

Inverse element: Each symmetry has an inverse: {\displaystyle \mathrm {id} }, the reflections {\displaystyle f_{\mathrm {h} }}{\displaystyle f_{\mathrm {v} }}{\displaystyle f_{\mathrm {d} }}{\displaystyle f_{\mathrm {c} }} and the 180° rotation {\displaystyle r_{2}} are their own inverse, because performing them twice brings the square back to its original orientation. The rotations {\displaystyle r_{3}} and {\displaystyle r_{1}} are each other’s inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table.

In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in {\displaystyle \mathrm {D} _{4}}, as, for example, {\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }} but {\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }}. In other words, {\displaystyle \mathrm {D} _{4}} is not abelian.

History

The modern concept of an abstract group developed out of several fields of mathematics.[9][10][11] The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois’s ideas were rejected by his contemporaries, and published only posthumously.[12][13] More general permutation groups were investigated in particular by Augustin Louis CauchyArthur Cayley‘s On the theory of groups, as depending on the symbolic equation {\displaystyle \theta ^{n}=1} (1854) gives the first abstract definition of a finite group.[14]

Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein‘s 1872 Erlangen program.[15] After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.[16]

The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss‘s number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker.[17] In 1847, Ernst Kummer made early attempts to prove Fermat’s Last Theorem by developing groups describing factorization into prime numbers.[18]

The convergence of these various sources into a uniform theory of groups started with Camille Jordan‘s Traité des substitutions et des équations algébriques (1870).[19] Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an “abstract group”, in the terminology of the time.[20] As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer‘s modular representation theory and Issai Schur‘s papers.[21] The theory of Lie groups, and more generally locally compact groups was studied by Hermann WeylÉlie Cartan and many others.[22] Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.[23]

The University of Chicago‘s 1960–61 Group Theory Year brought together group theorists such as Daniel GorensteinJohn G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research concerning this classification proof is ongoing.[24] Group theory remains a highly active mathematical branch,[b] impacting many other fields, as the examples below illustrate.

Elementary consequences of the group axioms

Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory.[25] For example, repeated applications of the associativity axiom show that the unambiguity of

{\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)}

generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted.[26]

Individual axioms may be “weakened” to assert only the existence of a left identity and left inverses. From these one-sided axioms, one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are no weaker.[27]

Uniqueness of identity element

The group axioms imply that the identity element is unique: If {\displaystyle e} and {\displaystyle f} are identity elements of a group, then {\displaystyle e=e\cdot f=f}. Therefore, it is customary to speak of the identity.[28]

Uniqueness of inverses

The group axioms also imply that the inverse of each element is unique: If a group element {\displaystyle a} has both {\displaystyle b} and {\displaystyle c} as inverses, then

 {\displaystyle b} {\displaystyle {}={}} {\displaystyle b\cdot e} since {\displaystyle e} is the identity element {\displaystyle {}={}} {\displaystyle b\cdot (a\cdot c)} since {\displaystyle c} is an inverse of {\displaystyle a}, so {\displaystyle e=a\cdot c} {\displaystyle {}={}} {\displaystyle (b\cdot a)\cdot c} by associativity, which allows rearranging the parentheses {\displaystyle {}={}} {\displaystyle e\cdot c} since {\displaystyle b} is an inverse of {\displaystyle a}, so {\displaystyle b\cdot a=e} {\displaystyle {}={}} {\displaystyle c} since {\displaystyle e} is the identity element.

Therefore, it is customary to speak of the inverse of an element.[28]

Division

Given elements {\displaystyle a} and {\displaystyle b} of a group {\displaystyle G}, there is a unique solution {\displaystyle x} in {\displaystyle G} to the equation {\displaystyle a\cdot x=b}, namely {\displaystyle a^{-1}\cdot b}. (One usually avoids using fraction notation {\displaystyle {\tfrac {b}{a}}} unless {\displaystyle G} is abelian, because of the ambiguity of whether it means {\displaystyle a^{-1}\cdot b} or {\displaystyle b\cdot a^{-1}}.)[29] It follows that for each {\displaystyle a} in {\displaystyle G}, the function {\displaystyle G\to G} that maps each {\displaystyle x} to {\displaystyle a\cdot x} is a bijection; it is called left multiplication by {\displaystyle a} or left translation by {\displaystyle a}.

Similarly, given {\displaystyle a} and {\displaystyle b}, the unique solution to {\displaystyle x\cdot a=b} is {\displaystyle b\cdot a^{-1}}. For each {\displaystyle a}, the function {\displaystyle G\to G} that maps each {\displaystyle x} to {\displaystyle x\cdot a} is a bijection called right multiplication by {\displaystyle a} or right translation by {\displaystyle a}.

Basic concepts

When studying sets, one uses concepts such as subset, function, and quotient by an equivalence relation. When studying groups, one uses instead subgroupshomomorphisms, and quotient groups. These are the appropriate analogues that take into account the existence of the group structure.[c]

Group homomorphisms

Group homomorphisms[d] are functions that respect group structure; they may be used to relate two groups. A homomorphism from a group {\displaystyle (G,\cdot )} to a group {\displaystyle (H,*)} is a function {\displaystyle \varphi :G\to H} such that

{\displaystyle \varphi (a\cdot b)=\varphi (a)*\varphi (b)} for all elements {\displaystyle a} and {\displaystyle b} in {\displaystyle G}.

It would be natural to require also that {\displaystyle \varphi } respect identities, {\displaystyle \varphi (1_{G})=1_{H}}, and inverses, {\displaystyle \varphi (a^{-1})=\varphi (a)^{-1}} for all {\displaystyle a} in {\displaystyle G}. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.[30]

The identity homomorphism of a group {\displaystyle G} is the homomorphism {\displaystyle \iota _{G}:G\to G} that maps each element of {\displaystyle G} to itself. An inverse homomorphism of a homomorphism {\displaystyle \varphi :G\to H} is a homomorphism {\displaystyle \psi :H\to G} such that {\displaystyle \psi \circ \varphi =\iota _{G}} and {\displaystyle \varphi \circ \psi =\iota _{H}}, that is, such that {\displaystyle \psi {\bigl (}\varphi (g){\bigr )}=g} for all {\displaystyle g} in {\displaystyle G} and such that {\displaystyle \varphi {\bigl (}\psi (h){\bigr )}=h} for all {\displaystyle h} in {\displaystyle H}. An isomorphism is a homomorphism that has an inverse homomorphism; equivalently, it is a bijective homomorphism. Groups {\displaystyle G} and {\displaystyle H} are called isomorphic if there exists an isomorphism {\displaystyle \varphi :G\to H}. In this case, {\displaystyle H} can be obtained from {\displaystyle G} simply by renaming its elements according to the function {\displaystyle \varphi }; then any statement true for {\displaystyle G} is true for {\displaystyle H}, provided that any specific elements mentioned in the statement are also renamed.

The collection of all groups, together with the homomorphisms between them, form a category, the category of groups.[31]

Subgroups

Informally, a subgroup is a group {\displaystyle H} contained within a bigger one, {\displaystyle G}: it has a subset of the elements of {\displaystyle G}, with the same operation.[32] Concretely, this means that the identity element of {\displaystyle G} must be contained in {\displaystyle H}, and whenever {\displaystyle h_{1}} and {\displaystyle h_{2}} are both in {\displaystyle H}, then so are {\displaystyle h_{1}\cdot h_{2}} and {\displaystyle h_{1}^{-1}}, so the elements of {\displaystyle H}, equipped with the group operation on {\displaystyle G} restricted to {\displaystyle H}, indeed form a group.

In the example of symmetries of a square, the identity and the rotations constitute a subgroup {\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}, highlighted in red in the group table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The subgroup test provides a necessary and sufficient condition for a nonempty subset H of a group G to be a subgroup: it is sufficient to check that {\displaystyle g^{-1}\cdot h\in H} for all elements {\displaystyle g} and {\displaystyle h} in {\displaystyle H}. Knowing a group’s subgroups is important in understanding the group as a whole.[e]

Given any subset {\displaystyle S} of a group {\displaystyle G}, the subgroup generated by {\displaystyle S} consists of products of elements of {\displaystyle S} and their inverses. It is the smallest subgroup of {\displaystyle G} containing {\displaystyle S}.[33] In the example of symmetries of a square, the subgroup generated by {\displaystyle r_{2}} and {\displaystyle f_{\mathrm {v} }} consists of these two elements, the identity element {\displaystyle \mathrm {id} }, and the element {\displaystyle f_{\mathrm {h} }=f_{\mathrm {v} }\cdot r_{2}}. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.

Cosets

In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup {\displaystyle H} determines left and right cosets, which can be thought of as translations of {\displaystyle H} by an arbitrary group element {\displaystyle g}. In symbolic terms, the left and right cosets of {\displaystyle H}, containing an element {\displaystyle g}, are

{\displaystyle gH=\{g\cdot h\mid h\in H\}} and {\displaystyle Hg=\{h\cdot g\mid h\in H\}}, respectively.[34]

The left cosets of any subgroup {\displaystyle H} form a partition of {\displaystyle G}; that is, the union of all left cosets is equal to {\displaystyle G} and two left cosets are either equal or have an empty intersection.[35] The first case {\displaystyle g_{1}H=g_{2}H} happens precisely when {\displaystyle g_{1}^{-1}\cdot g_{2}\in H}, i.e., when the two elements differ by an element of {\displaystyle H}. Similar considerations apply to the right cosets of {\displaystyle H}. The left cosets of {\displaystyle H} may or may not be the same as its right cosets. If they are (that is, if all {\displaystyle g} in {\displaystyle G} satisfy {\displaystyle gH=Hg}), then {\displaystyle H} is said to be a normal subgroup.

In {\displaystyle \mathrm {D} _{4}}, the group of symmetries of a square, with its subgroup {\displaystyle R} of rotations, the left cosets {\displaystyle gR} are either equal to {\displaystyle R}, if {\displaystyle g} is an element of {\displaystyle R} itself, or otherwise equal to {\displaystyle U=f_{\mathrm {c} }R=\{f_{\mathrm {c} },f_{\mathrm {d} },f_{\mathrm {v} },f_{\mathrm {h} }\}} (highlighted in green in the group table of {\displaystyle \mathrm {D} _{4}}). The subgroup {\displaystyle R} is normal, because {\displaystyle f_{\mathrm {c} }R=U=Rf_{\mathrm {c} }} and similarly for the other elements of the group. (In fact, in the case of {\displaystyle \mathrm {D} _{4}}, the cosets generated by reflections are all equal: {\displaystyle f_{\mathrm {h} }R=f_{\mathrm {v} }R=f_{\mathrm {d} }R=f_{\mathrm {c} }R}.)

Quotient groups

In some situations the set of cosets of a subgroup can be endowed with a group law, giving a quotient group or factor group. For this to be possible, the subgroup has to be normal. Given any normal subgroup N, the quotient group is defined by

{\displaystyle G/N=\{gN\mid g\in G\},}

where the notation {\displaystyle G/N} is read as “{\displaystyle G} modulo {\displaystyle N}“.[36] This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group {\displaystyle G}: the product of two cosets {\displaystyle gN} and {\displaystyle hN} is {\displaystyle (gN)\cdot (hN)=(gh)N} for all {\displaystyle g} and {\displaystyle h} in {\displaystyle G}. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map {\displaystyle G\to G/N} that associates to any element {\displaystyle g} its coset {\displaystyle gN} should be a group homomorphism, or by general abstract considerations called universal properties. The coset {\displaystyle eN=N} serves as the identity in this group, and the inverse of {\displaystyle gN} in the quotient group is {\displaystyle (gN)^{-1}=\left(g^{-1}\right)N}.[f]

Group table of the quotient group {\displaystyle \mathrm {D} _{4}/R}
{\displaystyle \cdot }{\displaystyle R}{\displaystyle U}
{\displaystyle R}{\displaystyle R}{\displaystyle U}
{\displaystyle U}{\displaystyle U}{\displaystyle R}

The elements of the quotient group {\displaystyle \mathrm {D} _{4}/R} are {\displaystyle R} itself, which represents the identity, and {\displaystyle U=f_{\mathrm {v} }R}. The group operation on the quotient is shown in the table. For example, {\displaystyle U\cdot U=f_{\mathrm {v} }R\cdot f_{\mathrm {v} }R=(f_{\mathrm {v} }\cdot f_{\mathrm {v} })R=R}. Both the subgroup {\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}, as well as the corresponding quotient are abelian, whereas {\displaystyle \mathrm {D} _{4}} is not abelian. Building bigger groups by smaller ones, such as {\displaystyle \mathrm {D} _{4}} from its subgroup {\displaystyle R} and the quotient {\displaystyle \mathrm {D} _{4}/R} is abstracted by a notion called semidirect product.

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group {\displaystyle \mathrm {D} _{4}}, for example, can be generated by two elements {\displaystyle r} and {\displaystyle f} (for example, {\displaystyle r=r_{1}}, the right rotation and {\displaystyle f=f_{\mathrm {v} }} the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

{\displaystyle r^{4}=f^{2}=(r\cdot f)^{2}=1,}

the group is completely described.[37] A presentation of a group can also be used to construct the Cayley graph, a device used to graphically capture discrete groups.[38]

Sub- and quotient groups are related in the following way: a subgroup {\displaystyle H} of {\displaystyle G} corresponds to an injective map {\displaystyle H\to G}, for which any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map {\displaystyle G\to G/N}.[g] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions. In general, homomorphisms are neither injective nor surjective. The kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.

Examples and applications

A periodic wallpaper pattern gives rise to a wallpaper group.
The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers.

Examples and applications of groups abound. A starting point is the group {\displaystyle \mathbb {Z} } of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra.

Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group.[39] By means of this connection, topological properties such as proximity and continuity translate into properties of groups.[h] For example, elements of the fundamental group are represented by loops. The second image shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.

In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background.[i] In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups.[40] Further branches crucially applying groups include algebraic geometry and number theory.[41]

In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in particular when implemented for finite groups.[42] Applications of group theory are not restricted to mathematics; sciences such as physicschemistry and computer science benefit from the concept.

Numbers

Many number systems, such as the integers and the rationals, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modulesvector spaces and algebras also form groups.

Integers

The group of integers {\displaystyle \mathbb {Z} } under addition, denoted {\displaystyle \left(\mathbb {Z} ,+\right)}, has been described above. The integers, with the operation of multiplication instead of addition, {\displaystyle \left(\mathbb {Z} ,\cdot \right)} do not form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, {\displaystyle a=2} is an integer, but the only solution to the equation {\displaystyle a\cdot b=1} in this case is {\displaystyle b={\tfrac {1}{2}}}, which is a rational number, but not an integer. Hence not every element of {\displaystyle \mathbb {Z} } has a (multiplicative) inverse.[j]

Rationals

The desire for the existence of multiplicative inverses suggests considering fractions

{\displaystyle {\frac {a}{b}}.}

Fractions of integers (with {\displaystyle b} nonzero) are known as rational numbers.[k] The set of all such irreducible fractions is commonly denoted {\displaystyle \mathbb {Q} }. There is still a minor obstacle for {\displaystyle \left(\mathbb {Q} ,\cdot \right)}, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no {\displaystyle x} such that {\displaystyle x\cdot 0=1}), {\displaystyle \left(\mathbb {Q} ,\cdot \right)} is still not a group.

However, the set of all nonzero rational numbers {\displaystyle \mathbb {Q} \smallsetminus \left\{0\right\}=\left\{q\in \mathbb {Q} \mid q\neq 0\right\}} does form an abelian group under multiplication, also denoted {\displaystyle \mathbb {Q} ^{\times }}.[l] Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of {\displaystyle a/b} is {\displaystyle b/a}, therefore the axiom of the inverse element is satisfied.

The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if division by other than zero is possible, such as in {\displaystyle \mathbb {Q} } – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.[m]

Modular arithmetic

The hours on a clock form a group that uses addition modulo 12. Here, 9 + 4 ≡ 1.

Modular arithmetic for a modulus {\displaystyle n} defines any two elements {\displaystyle a} and {\displaystyle b} that differ by a multiple of {\displaystyle n} to be equivalent, denoted by {\displaystyle a\equiv b{\pmod {n}}}. Every integer is equivalent to one of the integers from {\displaystyle 0} to {\displaystyle n-1}, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent representative. Modular addition, defined in this way for the integers from {\displaystyle 0} to {\displaystyle n-1}, forms a group, denoted as {\displaystyle \mathrm {Z} _{n}} or {\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}, with {\displaystyle 0} as the identity element and {\displaystyle n-a} as the inverse element of {\displaystyle a}.

A familiar example is addition of hours on the face of a clock, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on {\displaystyle 9} and is advanced {\displaystyle 4} hours, it ends up on {\displaystyle 1}, as shown in the illustration. This is expressed by saying that {\displaystyle 9+4} is congruent to {\displaystyle 1} “modulo {\displaystyle 12}” or, in symbols,

{\displaystyle 9+4\equiv 1{\pmod {12}}.}

For any prime number {\displaystyle p}, there is also the multiplicative group of integers modulo {\displaystyle p}.[43] Its elements can be represented by {\displaystyle 1} to {\displaystyle p-1}. The group operation, multiplication modulo {\displaystyle p}, replaces the usual product by its representative, the remainder of division by {\displaystyle p}. For example, for {\displaystyle p=5}, the four group elements can be represented by {\displaystyle 1,2,3,4}. In this group, {\displaystyle 4\cdot 4\equiv 1{\bmod {5}}}, because the usual product {\displaystyle 16} is equivalent to {\displaystyle 1}: when divided by {\displaystyle 5} it yields a remainder of {\displaystyle 1}. The primality of {\displaystyle p} ensures that the usual product of two representatives is not divisible by {\displaystyle p}, and therefore that the modular product is nonzero.[n] The identity element is represented by {\displaystyle 1}, and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer {\displaystyle a} not divisible by {\displaystyle p}, there exists an integer {\displaystyle b} such that

{\displaystyle a\cdot b\equiv 1{\pmod {p}},}

that is, such that {\displaystyle p} evenly divides {\displaystyle a\cdot b-1}. The inverse {\displaystyle b} can be found by using Bézout’s identity and the fact that the greatest common divisor {\displaystyle \gcd(a,p)} equals {\displaystyle 1}.[44] In the case {\displaystyle p=5} above, the inverse of the element represented by {\displaystyle 4} is that represented by {\displaystyle 4}, and the inverse of the element represented by {\displaystyle 3} is represented by {\displaystyle 2}, as {\displaystyle 3\cdot 2=6\equiv 1{\bmod {5}}}. Hence all group axioms are fulfilled. This example is similar to {\displaystyle \left(\mathbb {Q} \smallsetminus \left\{0\right\},\cdot \right)} above: it consists of exactly those elements in the ring {\displaystyle \mathbb {Z} /p\mathbb {Z} } that have a multiplicative inverse.[45] These groups, denoted {\displaystyle \mathbb {F} _{p}^{\times }}, are crucial to public-key cryptography.[o]

Cyclic groups

The 6th complex roots of unity form a cyclic group. {\displaystyle z} is a primitive element, but {\displaystyle z^{2}} is not, because the odd powers of {\displaystyle z} are not a power of {\displaystyle z^{2}}.

cyclic group is a group all of whose elements are powers of a particular element {\displaystyle a}.[46] In multiplicative notation, the elements of the group are

{\displaystyle \dots ,a^{-3},a^{-2},a^{-1},a^{0},a,a^{2},a^{3},\dots ,}

where {\displaystyle a^{2}} means {\displaystyle a\cdot a}{\displaystyle a^{-3}} stands for {\displaystyle a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1}}, etc.[p] Such an element {\displaystyle a} is called a generator or a primitive element of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as

{\displaystyle \dots ,(-a)+(-a),-a,0,a,a+a,\dots .}

In the groups {\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)} introduced above, the element {\displaystyle 1} is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are {\displaystyle 1}. Any cyclic group with {\displaystyle n} elements is isomorphic to this group. A second example for cyclic groups is the group of {\displaystyle n}th complex roots of unity, given by complex numbers {\displaystyle z} satisfying {\displaystyle z^{n}=1}. These numbers can be visualized as the vertices on a regular {\displaystyle n}-gon, as shown in blue in the image for {\displaystyle n=6}. The group operation is multiplication of complex numbers. In the picture, multiplying with {\displaystyle z} corresponds to a counter-clockwise rotation by 60°.[47] From field theory, the group {\displaystyle \mathbb {F} _{p}^{\times }} is cyclic for prime {\displaystyle p}: for example, if {\displaystyle p=5}{\displaystyle 3} is a generator since {\displaystyle 3^{1}=3}{\displaystyle 3^{2}=9\equiv 4}{\displaystyle 3^{3}\equiv 2}, and {\displaystyle 3^{4}\equiv 1}.

Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element {\displaystyle a}, all the powers of {\displaystyle a} are distinct; despite the name “cyclic group”, the powers of the elements do not cycle. An infinite cyclic group is isomorphic to {\displaystyle (\mathbb {Z} ,+)}, the group of integers under addition introduced above.[48] As these two prototypes are both abelian, so are all cyclic groups.

The study of finitely generated abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.[49]

Symmetry groups

The (2,3,7) triangle group, a hyperbolic reflection group, acts on this tiling of the hyperbolic plane[50]

Symmetry groups are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below).[51] Conceptually, group theory can be thought of as the study of symmetry.[q] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element can be associated to some operation on X and the composition of these operations follows the group law. For example, an element of the (2,3,7) triangle group acts on a triangular tiling of the hyperbolic plane by permuting the triangles.[50] By a group action, the group pattern is connected to the structure of the object being acted on.

In chemical fields, such as crystallographyspace groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties.[52] For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.[53]

Group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.[54]

Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons.[55]

 displaysicosahedral symmetry[56] Ammonia, NH3. Its symmetry group is of order 6, generated by a 120° rotation and a reflection.[57] Cubane C8H8 featuresoctahedral symmetry.[58] The tetrachloroplatinate(II) ion, [PtCl4]2- exhibits square-planar geometry

Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players.[59] Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.[r] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.[60General linear group and representation theory

ectors (the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the {\displaystyle x}-coordinate by factor 2.

Matrix groups consist of matrices together with matrix multiplication. The general linear group {\displaystyle \mathrm {GL} (n,\mathbb {R} )} consists of all invertible {\displaystyle n}-by-{\displaystyle n} matrices with real entries.[61] Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group {\displaystyle \mathrm {SO} (n)}. It describes all possible rotations in {\displaystyle n} dimensions. Rotation matrices in this group are used in computer graphics.[62]

Representation theory is both an application of the group concept and important for a deeper understanding of groups.[63][64] It studies the group by its group actions on other spaces. A broad class of group representations are linear representations in which the group acts on a vector space, such as the three-dimensional Euclidean space {\displaystyle \mathbb {R} ^{3}}. A representation of a group {\displaystyle G} on an {\displaystyle n}dimensional real vector space is simply a group homomorphism {\displaystyle \rho :G\to \mathrm {GL} (n,\mathbb {R} )} from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.[s]

A group action gives further means to study the object being acted on.[t] On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups.[63][65]

Galois groups

Galois groups were developed to help solve polynomial equations by capturing their symmetry features.[66][67] For example, the solutions of the quadratic equation {\displaystyle ax^{2}+bx+c=0} are given by

{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}

Each solution can be obtained by replacing the {\displaystyle \pm } sign by {\displaystyle +} or {\displaystyle -}; analogous formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher.[68] In the quadratic formula, changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomials and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their solvability) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and roots similar to the formula above.[69]

Modern Galois theory generalizes the above type of Galois groups by shifting to field theory and considering field extensions formed as the splitting field of a polynomial. This theory establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.[70]

Finite groups

A group is called finite if it has a finite number of elements. The number of elements is called the order of the group.[71] An important class is the symmetric groups {\displaystyle \mathrm {S} _{N}}, the groups of permutations of {\displaystyle N} objects. For example, the symmetric group on 3 letters {\displaystyle \mathrm {S} _{3}} is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (factorial of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group {\displaystyle \mathrm {S} _{N}} for a suitable integer {\displaystyle N}, according to Cayley’s theorem. Parallel to the group of symmetries of the square above, {\displaystyle \mathrm {S} _{3}} can also be interpreted as the group of symmetries of an equilateral triangle.

The order of an element {\displaystyle a} in a group {\displaystyle G} is the least positive integer {\displaystyle n} such that {\displaystyle a^{n}=e}, where {\displaystyle a^{n}} represents

{\displaystyle \underbrace {a\cdots a} _{n{\text{ factors}}},}

that is, application of the operation “{\displaystyle \cdot }” to {\displaystyle n} copies of {\displaystyle a}. (If “{\displaystyle \cdot }” represents multiplication, then {\displaystyle a^{n}} corresponds to the {\displaystyle n}th power of {\displaystyle a}.) In infinite groups, such an {\displaystyle n} may not exist, in which case the order of {\displaystyle a} is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.

More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups: Lagrange’s Theorem states that for a finite group {\displaystyle G} the order of any finite subgroup {\displaystyle H} divides the order of {\displaystyle G}. The Sylow theorems give a partial converse.

The dihedral group {\displaystyle \mathrm {D} _{4}} of symmetries of a square is a finite group of order 8. In this group, the order of {\displaystyle r_{1}} is 4, as is the order of the subgroup {\displaystyle R} that this element generates. The order of the reflection elements {\displaystyle f_{\mathrm {v} }} etc. is 2. Both orders divide 8, as predicted by Lagrange’s theorem. The groups {\displaystyle \mathbb {F} _{p}^{\times }} of multiplication modulo a prime {\displaystyle p} have order {\displaystyle p-1}.

Finite abelian groups

Any finite abelian group is isomorphic to a product of finite cyclic groups; this statement is part of the fundamental theorem of finitely generated abelian groups.

Any group of prime order {\displaystyle p} is isomorphic to the cyclic group {\displaystyle \mathrm {Z} _{p}} (a consequence of Lagrange’s theorem). Any group of order {\displaystyle p^{2}} is abelian, isomorphic to {\displaystyle \mathrm {Z} _{p^{2}}} or {\displaystyle \mathrm {Z} _{p}\times \mathrm {Z} _{p}}. But there exist nonabelian groups of order {\displaystyle p^{3}}; the dihedral group {\displaystyle \mathrm {D} _{4}} of order {\displaystyle 2^{3}} above is an example.[72]

Simple groups

When a group {\displaystyle G} has a normal subgroup {\displaystyle N} other than {\displaystyle \{1\}} and {\displaystyle G} itself, questions about {\displaystyle G} can sometimes be reduced to questions about {\displaystyle N} and {\displaystyle G/N}. A nontrivial group is called simple if it has no such normal subgroup. Finite simple groups are to finite groups as prime numbers are to positive integers: they serve as building blocks, in a sense made precise by the Jordan–Hölder theorem.

Classification of finite simple groups

Computer algebra systems have been used to list all groups of order up to 2000.[u] But classifying all finite groups is a problem considered too hard to be solved.

The classification of all finite simple groups was a major achievement in contemporary group theory. There are several infinite families of such groups, as well as 26 “sporadic groups” that do not belong to any of the families. The largest sporadic group is called the monster group. The monstrous moonshine conjectures, proved by Richard Borcherds, relate the monster group to certain modular functions.[73]

The gap between the classification of simple groups and the classification of all groups lies in the extension problem.[74]

An equivalent definition of group consists of replacing the “there exist” part of the group axioms by operations whose result is the element that must exist. So, a group is a set {\displaystyle G} equipped with a binary operation {\displaystyle G\times G\rightarrow G} (the group operation), a unary operation {\displaystyle G\rightarrow G} (which provides the inverse) and a nullary operation, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids existential quantifiers and is used in computing with groups and for computer-aided proofs.

This way of defining groups lends itself to generalizations such as the notion of a group objects in a category. Briefly this is an object (that is, examples of another mathematical structure) which comes with transformations (called morphisms) that mimic the group axioms.[75]

Topological groups

The unit circle in the complex plane under complex multiplication is a Lie group and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold and thus a Lie group, because every small piece, such as the red arc in the figure, looks like a part of the real line (shown at the bottom).

Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally, {\displaystyle g\cdot h} and {\displaystyle g^{-1}} must not vary wildly if {\displaystyle g} and {\displaystyle h} vary only a little. Such groups are called topological groups, and they are the group objects in the category of topological spaces.[76] The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other topological field, such as the field of complex numbers or the field of p-adic numbers. These examples are locally compact, so they have Haar measures and can be studied via harmonic analysis. Other locally compact topological groups include the group of points of an algebraic group over a local field or adele ring; these are basic to number theory[77] Galois groups of infinite algebraic field extensions are equipped with the Krull topology, which plays a role in infinite Galois theory.[78] A generalization used in algebraic geometry is the étale fundamental group.[79]

Lie groups

Lie group is a group that also has the structure of a differentiable manifold; informally, this means that it looks locally like a Euclidean space of some fixed dimension.[80] Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be smooth.

A standard example is the general linear group introduced above: it is an open subset of the space of all {\displaystyle n}-by-{\displaystyle n} matrices, because it is given by the inequality

{\displaystyle \det(A)\neq 0,}

where {\displaystyle A} denotes an {\displaystyle n}-by-{\displaystyle n} matrix.[81]

Lie groups are of fundamental importance in modern physics: Noether’s theorem links continuous symmetries to conserved quantities.[82] Rotation, as well as translations in space and time, are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.[v] Another example is the group of Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of spacetime in special relativity.[83] The full symmetry group of Minkowski space, i.e., including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories.[84] Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory. An important example of a gauge theory is the Standard Model, which describes three of the four known fundamental forces and classifies all known elementary particles.[85]

Generalizations

Group-like structures
TotalityαAssociativityIdentityDivisionComm
SemigroupoidUnneededRequiredUnneededUnneededUnnee
Small categoryUnneededRequiredRequiredUnneededUnnee
GroupoidUnneededRequiredRequiredRequiredUnnee
MagmaRequiredUnneededUnneededUnneededUnneeded
QuasigroupRequiredUnneededUnneededRequiredUnnee
Unital magmaRequiredUnneededRequiredUnneededUnnee
SemigroupRequiredRequiredUnneededUnneededUnnee
LoopRequiredUnneededRequiredRequiredUnnee
MonoidRequiredRequiredRequiredUnneededUnnee
GroupRequiredRequiredRequiredRequiredUnnee
Commutative monoidRequiredRequiredRequiredUnneededRequired
Abelian groupRequiredRequiredRequiredRequiredRequir
The closure axiom, used by many sources and defined differently, is equivalent.

In abstract algebra, more general structures are defined by relaxing some of the axioms defining a group.[31][86][87] For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers {\displaystyle \mathbb {N} } (including zero) under addition form a monoid, as do the nonzero integers under multiplication {\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}, see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as {\displaystyle (\mathbb {Q} \smallsetminus \{0\},\cdot )} is derived from {\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}, known as the Grothendieck groupGroupoids are similar to groups except that the composition {\displaystyle a\cdot b} need not be defined for all {\displaystyle a} and {\displaystyle b}. They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e., an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n-ary group.[88] The table gives a list of several structures generalizing groups.