The manipulations of this Rubik's Cube form the Rubik's Cube group.

In mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element also in the set while satisfying four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation; the addition of any two integers forms another integer. The abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas 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. For example, a symmetry group encodes symmetry features of a geometrical object: the group 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. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity.

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 announced in 1983.^aa[›] Since the 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] together with addition.

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. Thus, adding two integers never yields some other type of number, such as a fraction. 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.

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

Definition [

A group is a set, G, together with an operation • (called the group law of G) that combines any two elements a and b to form another element, denoted a • b or ab. 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, (a • b) • c = a • (b • c).

Identity element

There exists an element e in G, such that for every element a in G, the equation e • a = a • e = a holds. Such an element is unique (see below), and thus one speaks of the identity element.

Inverse element

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

The result of an operation may depend on the order of the operands. 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 always holds in the group of integers under addition, because a + b = b + a for any two integers (commutativity of addition). Groups for which the commutativity equation a • b = b • a always holds are called abelian groups (in honor of Niels Abel). The symmetry group described in the following section is an example of a group that is not abelian.

The identity element of a group G is often written as 1 or 1_G,^[5] a notation inherited from the multiplicative identity. The identity element may also be written as 0, especially if the group operation is denoted by +, in which case the group is called an additive group. The identity element can also be written as id.

The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G of the group (G, •)" or "an element of the underlying set G of the group (G, •)". Usually, it is clear from the context whether a symbol like G refers to a group or to an underlying set.

Second example: a symmetry group [

Two figures in the plane are congruent if one can be changed into the other using a combination of rotations, reflections, 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:

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 to distinguish between them.

• the identity operation leaving everything unchanged, denoted id;
• rotations of the square around its center 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).

These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r₁ sends a point to its rotation 90° right around the square's center, and f_h sends a point to its reflection across the square's vertical middle line. Composing two of these symmetry functions gives another symmetry function. These symmetries determine a group called the dihedral group of degree 4 and denoted D₄. The underlying set of the group is the above set of symmetry functions, and the group operation is function composition.^[6] 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 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 is the same notation that is used for 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: Starting with three elements a, b and c of D₄, 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 a and b into a single symmetry, then to compose that symmetry with c. The other way is to first compose b and c, then to compose the resulting symmetry with a. The associativity condition

   (a • b) • c = a • (b • c)

   means that these two ways are the same, i.e., a product of many group elements can be simplified in any order. 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.

   While associativity is true for the symmetries of the square and addition of numbers, it is not true for all operations. For instance, subtraction of numbers is not associative: (7 − 3) − 2 = 2 is not the same as 7 − (3 − 2) = 6.

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 the following transformations—identity id, the flips f_h, f_v, f_d, f_c and the 180° rotation r₂—is its own inverse, because performing it twice brings the square back to its original orientation. The rotations r₃ and r₁ 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. 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.^[7][8][9] 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.^[10][11] 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.^[12]

Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program.^[13] 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.^[14]

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.^[15] In 1847, Ernst Kummer led early attempts to prove Fermat's Last Theorem to a climax by developing groups describing factorization into prime numbers.^[16]

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).^[17] Walther von Dyck (1882) gave the first statement of the modern definition of an abstract group.^[18] 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.^[19] The theory of Lie groups, and more generally locally compact groups was pushed by Hermann Weyl, Élie Cartan and many others.^[20] 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.^[21]

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.^[22] These days, group theory is still a highly active mathematical branch crucially impacting many other fields.^a[›]

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.[23] For example, repeated applications of the associativity axiom show that thee unambiguity of

a • b • c = (a • b) • c = a • (b • c)

generalizes to more than three factors. Because this implies that pSource: this wikipedia article, under CC-BY-SA.