The possible manipulations of this Rubik's Cube form a group.

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity and invertibility. While these are familiar from many mathematical structures, such as number systems—for example, the integers endowed with the addition operation form a group—the formulation of the axioms is detached from the concrete nature of the group and its operation. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of groups in numerous areas—both within and outside mathematics—makes them a central organizing principle of contemporary mathematics.^[1][2]

Groups share a fundamental kinship with the notion of symmetry. A symmetry group encodes symmetry features of a geometrical object: it consists of the set of transformations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other. Such symmetry groups, particularly the continuous Lie groups, play an important role in many academic disciplines. Matrix groups, for example, can be used to understand fundamental physical laws underlying special relativity and symmetry phenomena in molecular chemistry.

The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—a very active mathematical discipline—studies groups in their own right.^a[›] To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient 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 (its group representations), both from a theoretical and a computational point of view. A particularly rich theory has been developed for finite groups, which culminated with the monumental classification of finite simple groups completed in 1983. Since mid-1980s geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.

[ Definition and illustration

[ First example: the integers

One of the most familiar groups is the set of integers Z which consists of the numbers

..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...^[3]

The following properties of integer addition serve as a model for the abstract group axioms given in the definition below.

1. For any two integers a and b, the sum a + b is also an integer. In other words, the process of adding integers two at a time can never yield a result that is not an integer. This property is known as closure under addition.
2. For all integers a, b and c, (a + b) + c = a + (b + c). Expressed in words, adding a to b first, and then adding the result to c gives the same final result as adding a to the sum of b and c, a property known as associativity.
3. If a is any integer, then 0 + a = a + 0 = a. Zero is called the identity element of addition because adding it to any integer returns the same integer.
4. For every integer a, there is an integer b such that a + b = b + a = 0. The integer b is called the inverse element of the integer a and is denoted −a.

[ Definition

The integers, together with the operation "+", form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures without dealing with every concrete case separately, the following abstract definition is developed to encompass the above example along with many others, one of which is the symmetry group detailed below.

A group is a set, G, together with an operation "•" that combines any two elements a and b to form another element denoted a • b. The symbol "•" is a general placeholder for a concretely given operation, such as the addition above. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms:^[4]

Closure

For all a, b in G, the result of the operation a • b is also in G.^b[›]

Associativity

For all a, b and c in G, the equation (a • b) • c = a • (b • c) holds.

Identity element

There exists an element e in G, such that for all elements a in G, the equation e • a = a • e = a holds.

Inverse element

For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element.

The order in which the group operation is carried out can be significant. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation

a • b = b • a

may not always be true. This equation does always hold in the group of integers under addition, because a + b = b + a for any two integers (commutativity of addition). However, it does not always hold in the symmetry group below. Groups for which the equation a • b = b • a always holds are called abelian (in honor of Niels Abel). Thus, the integer addition group is abelian, but the following symmetry group is not.

[ Second example: a symmetry group

The symmetries (i.e., rotations and reflections) of a square form a group called a dihedral group, and denoted D₄.^[5] The following symmetries occur:

id (keeping it as is)	r1 (rotation by 90° right)	r2 (rotation by 180° right)	r3 (rotation by 270° right)
fv (vertical flip)	fh (horizontal flip)	fd (diagonal flip)	fc (counter-diagonal flip)
The elements of the symmetry group of the square (D4). The vertices are colored and numbered only to visualize the operations.

• the identity operation leaving everything unchanged, denoted id;
• rotations of the square by 90° right, 180° right, and 270° right, denoted by r₁, r₂ and r₃, respectively;
• reflections about the vertical and horizontal middle line (f_h and f_v), or through the two diagonals (f_d and f_c).

Any two symmetries a and b can be composed; i.e., applied one after another. The result of performing first a and then b is written symbolically from right to left as

b • a ("apply the symmetry b after performing the symmetry a". The right-to-left notation stems from composition of functions).

The group table on the right lists the results of all such compositions possible. For example, rotating by 270° right (r₃) and then flipping horizontally (f_h) is the same as performing a reflection along the diagonal (f_d). Using the above symbols, highlighted in blue in the group table:

f_h • r₃ = f_d.

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

1. The closure axiom demands that the composition b • a of any two symmetries a and b is also a symmetry. Another example for the group operation is

   r₃ • f_h = f_c,

   i.e. rotating 270° right after flipping horizontally equals flipping along the counter-diagonal (f_c). Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table.
2. The associativity constraint deals with composing more than two symmetries: given three elements a, b and c of D₄, there are two possible ways of computing "a then b then c". The requirement

   (a • b) • c = a • (b • c)

   means that the composition of the three elements is independent of the priority of the operations, i.e. composing first a after b, and c to the result a • b thereof amounts to performing a after the composition of b and c. For example, (f_d • f_v) • r₂ = f_d • (f_v • r₂) can be checked using the group table at the right

   (fd • fv) • r2	=	r3 • r2	=	r1, which equals
   fd • (fv • r2)	=	fd • fh	=	r1.

3. The identity element is the symmetry id leaving everything unchanged: for any symmetry a, performing id after a (or a after id) equals a, in symbolic form,

   id • a = a,

   a • id = a.

4. An inverse element undoes the transformation of some other element. Every symmetry can be undone: each of transformations—identity id, the flips f_h, f_v, f_d, f_c and the 180° rotation r₂—is its own inverse, because performing each one twice brings the square back to its original orientation. The rotations r₃ and r₁ are each other's inverse, because rotating one way and then by the same angle the other way leaves the square unchanged. In symbols,

   f_h • f_h = id,

   r₃ • r₁ = r₁ • r₃ = id.

In contrast to the group of integers above, where the order of the operation is irrelevant, it does matter in D₄: f_h • r₁ = f_c but r₁ • f_h = f_d. In other words, D₄ is not abelian, which makes the group structure more difficult than the integers introduced first.

[ History

The modern concept of an abstract group developed out of several fields of mathematics.^[6][7][8] 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' ideas were rejected by his contemporaries, and published only posthumously.^[9][10] More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θⁿ = 1 (1854) gives the first abstract definition of a finite group.^[11]

Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program.^[12] 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.^[13]

The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker.^[14] In 1847, Ernst Kummer led early attempts to prove Fermat's Last Theorem to a climax by developing groups describing factorization into prime numbers.^[15]

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).^[16] Walther von Dyck (1882) gave the first statement of the modern definition of an abstract group.^[17] 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.^[18] The theory of Lie groups, and more generally locally compact groups was pushed by Hermann Weyl, Élie Cartan and many others.^[19] Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by pivotal work of Armand Borel and Jacques Tits.^[20]

The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, classified all finite simple groups in 1982. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification.^[21] These days, group theory is still a highly active mathematical branch crucially impacting many other fields.^a[›]

[ Simple 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.^[22] For example, repeated applications of the associativity axiom show that the unambiguity of

a • b • c = (a • b) • c = a • (b • 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.^[23]

The axioms may be weakened to assert only the existence of a left identity and left inverses. Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one given above.^[24]

[ Uniqueness of identity element and inverses

Two important consequences of the group axioms are the uniqueness of the identity element and the uniqueness of inverse elements. There can be only one identity element in a group, and each element in a group has exactly one inverse element. Thus, it is customary to speak of the identity, and the inverse of an element.[25]

To prove the uniqueness of an inverse element of a, suppose that a has two inverses, denoted l and r. Then

l	=	l • e		as e is the identity element
	=	l • (a • r)		because r is an inverse of a, so e = a • r
	=	(l • a) • r		by associativity, which allows to rearrange the parentheses
	=	e • r		since l is an inverse of a, i.e. l • a = e
	=	r		for e is the identity element

Hence the two extremal terms l and r are connected by a chain of equalities, so they agree. In other words there is only one inverse element of a.

[ Division

In groups, it is possible to perform division: given elements a and b of the group G, there is exactly one solution x in G to the equation x • a = b.^[25] In fact, right multiplication of the equation by a⁻¹ gives the solution x = x • a • a⁻¹ = b • a⁻¹. Similarly there is exactly one solution y in G to the equation a • y = b, namely y = a⁻¹ • b. In general, x and y need not agree.

[ Basic concepts

To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.^c[›] There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which for example sets—being "structureless"—don't have) constructions related to groups have to be compatible with the group operation. This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. The principle of "preserving structures"—a recurring topic in mathematics throughout—is an instance of working in a category, in this case the category of groups.^[26]

[ Group homomorphisms

Group homomorphisms^g[›] are functions that preserve group structure. A function a: G → H between two groups is a homomorphism if the equation

a(g • k) = a(g) • a(k).

holds for all elements g, k in G, i.e. the result is the same when performing the group operation after or before applying the map a. This requirement ensures that a(1_G) = 1_H, and also a(g)⁻¹ = a(g⁻¹) for all g in G. Thus a group homomorphism respects all the structure of G provided by the group axioms.^[27]

Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another (in each of the two possible orders) equal the identity function of G and H, respectively. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of view, isomorphic groups carry the same information. For example, proving that g • g = 1 for some element g of G is equivalent to proving that a(g) • a(g) = 1, because applying a to the first equality yields the second, and applying b to the second gives back the first.

[ Subgroups

Informally, a subgroup is a group H contained within a bigger one, G.^[28] Concretely, the identity element of G is contained in H, and whenever h₁ and h₂ are in H, then so are h₁ • h₂ and h₁⁻¹, so the elements of H, equipped with the group operation on G restricted to H, form indeed a group.

In the example above, the identity and the rotations constitute a subgroup R = {id, r₁, r₂, r₃}, highlighted in red in the group table above: 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° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a subset H of a group G to be a subgroup: it is sufficient to check that g⁻¹h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.^d[›]

Given any subset S of a group G, the subgroup generated by S consists of products of elements of S and their inverses. It is the smallest subgroup of G containing S.^[29] In the introductory example above, the subgroup generated by r₂ and f_v consists of these two elements, the identity element id and f_h = f_v • r₂. 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 D₄ above, once a flip is performed, the square never gets back to the r₂ configuration by just applying the rotation operations (and no further flips), i.e. the rotation operations are irrelevant to the question whether a flip has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right coset of H containing g are

gH = {gh, h ∈ H} and Hg = {hg, h ∈ H}, respectively.^[30]

The cosets of any subgroup H form a partition of G; that is, the union of all left cosets is equal to G and two left cosets are either equal or have an empty intersection.^[31] The first case g₁H = g₂H happens precisely when g₁⁻¹g₂ ∈ H, i.e. if the two elements differ by an element of H. Similar considerations apply to the right cosets of H. The left and right cosets of H may or may not be equal. If they are, i.e. for all g in G, gH = Hg, then H is said to be a normal subgroup. One may then simply refer to N as the set of cosets.

In D₄, the introductory symmetry group, the left cosets gR of the subgroup R consisting of the rotations are either equal to R, if g is an element of R itself, or otherwise equal to U = f_vR = {f_v, f_d, f_h, f_c} (highlighted in green). The subgroup R is also normal, because f_vR = U = Rf_v and similarly for any element other than f_v.

[ Quotient groups

In addition to disregarding the internal structure of a subgroup by considering its cosets, it is desirable to endow this coarser entity with a group law called 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

G / N = {gN, g ∈ G}, "G modulo N".^[32]

This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to any element g its coset gN be a group homomorphism, or by general abstract considerations called universal properties. The coset eN = N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)⁻¹ = (g⁻¹)N.^e[›]

The elements of the quotient group D₄ / R are R itself, which represents the identity, and U = f_vR. The group operation on the quotient is shown at the right. For example, U • U = f_vR • f_vR = (f_v • f_v)R = R. Both the subgroup R = {id, r₁, r₂, r₃}, as well as the corresponding quotient are abelian, whereas D₄ is not abelian. Building bigger groups by smaller ones, such as D₄ from its subgroup R and the quotient D₄ / R is abstracted by a notion called semidirect product.

Quotient 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 D₄, for example, can be generated by two elements r and f (for example, r = r₁, the right rotation and f = f_v the vertical (or any other) flip), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

r ⁴ = f ² = (rf)² = 1,^[33]

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

Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. 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 G → G / N.^y[›] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. 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 thee group Z of integers with addition as group operation, introduced above. If instead of addition multiplicaSource: this wikipedia article, under GFDL.