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 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]
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. 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.
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, ...The following properties of integer addition serve as a model for the abstract group axioms given in the definition below.
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 . 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, , must satisfy four requirements known as the group axioms:
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 • amay 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.
The symmetries (i.e., rotations and reflections) of a square form a group called a dihedral group, and denoted D4. 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. |
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 (fh) is the same as performing a reflection along the diagonal (fd). Using the above symbols, highlighted in blue in the group table:
fh • r3 = fd.
| • | id | r1 | r2 | r3 | fv | fh | fd | fc |
|---|---|---|---|---|---|---|---|---|
| id | id | r1 | r2 | r3 | fv | fh | fd | fc |
| r1 | r1 | r2 | r3 | id | fc | fd | fv | fh |
| r2 | r2 | r3 | id | r1 | fh | fv | fc | fd |
| r3 | r3 | id | r1 | r2 | fd | fc | fh | fv |
| fv | fv | fd | fh | fc | id | r2 | r1 | r3 |
| fh | fh | fc | fv | fd | r2 | id | r3 | r1 |
| fd | fd | fh | fc | fv | r3 | r1 | id | r2 |
| fc | fc | fv | fd | fh | r1 | r3 | r2 | id |
| The elements id, r1, r2, and r3 form a subgroup, highlighted in red (upper left region). A left and right coset of this subgroup is highlighted in green (in the last row) and yellow (last column), respectively. | ||||||||
r3 • fh = fc,i.e. rotating 270° right after flipping horizontally equals flipping along the counter-diagonal (fc). Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table.
(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 thereof amounts to performing a after the composition of b and c.For example, (fd • fv) • r2 = fd • (fv • 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. |
id • a = a,
a • id = a.
fh • fh = 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: fh • r1 = fc but r1 • fh = fd. In other words, D4 is not abelian, which makes the group structure more difficult than the integers introduced first.
See main article: History of group theory. The modern concept of an abstract group developed out of several fields of mathematics. 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. 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.
Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. 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.
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. In 1847, Ernst Kummer led early attempts to prove Fermat's Last Theorem to a climax by developing groups describing factorization into prime numbers.
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). Walther von Dyck (1882) gave the first statement of the modern definition of an abstract group. 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. The theory of Lie groups, and more generally locally compact groups was pushed by Hermann Weyl, Élie Cartan and many others. 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.
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. These days, group theory is still a highly active mathematical branch crucially impacting many other fields.
See main article: Elementary group theory. Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory. 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.
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.
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.
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.
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. 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.
The following sections use mathematical symbols such as X = to denote a set X containing elements x, y, and z, or alternatively x ∈ X to restate that x is an element of X. The notation means f is a function assigning to every element of X an element of Y.
See also: Glossary of group theory. To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed. 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.
See main article: Group homomorphism. Group homomorphisms 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(1G) = 1H, 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.
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.
See main article: Subgroup. Informally, a subgroup is a group H contained within a bigger one, G. Concretely, the identity element of G is contained in H, and whenever h1 and h2 are in H, then so are 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 =