In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. 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; 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 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 subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory 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; 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. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, 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 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically 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, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem 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. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; 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, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the 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, 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 is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, fields. Speaking, it is the property of having no infinitely large or infinitely small elements, it was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder. The notion arose from the theory of magnitudes of Ancient Greece. An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has a pair of non-zero elements, one of, infinitesimal with respect to the other, is said to be non-Archimedean. For example, a linearly ordered group, Archimedean is an Archimedean group; this can be made precise in various contexts with different formulations. For example, in the context of ordered fields, one has the axiom of Archimedes which formulates this property, where the field of real numbers is Archimedean, but that of rational functions in real coefficients is not.
The concept was named by Otto Stolz after the ancient Greek geometer and physicist Archimedes of Syracuse. The Archimedean property appears in Book V of Euclid's Elements as Definition 4: Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another; because Archimedes credited it to Eudoxus of Cnidus it is known as the "Theorem of Eudoxus" or the Eudoxus axiom. Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs. Let x and y be positive elements of a linearly ordered group G. X is infinitesimal with respect to y if, for every natural number n, the multiple nx is less than y, that is, the following inequality holds: x + ⋯ + x ⏟ n terms < y. The group G is Archimedean. Additionally, if K is an algebraic structure with a unit — for example, a ring — a similar definition applies to K. If x is infinitesimal with respect to 1 x is an infinitesimal element. If y is infinite with respect to 1 y is an infinite element.
The algebraic structure K is Archimedean if it has no infinite elements and no infinitesimal elements. An ordered field has some additional properties. One may assume. If x is infinitesimal 1/x is infinite, vice versa. Therefore, to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements. If x is infinitesimal and r is a rational number r x is infinitesimal; as a result, given a general element c, the three numbers c/2, c, 2c are either all infinitesimal or all non-infinitesimal. In this setting, an ordered field K is Archimedean when the following statement, called the axiom of Archimedes, holds: Let x be any element of K. There exists a natural number n such that n > x. Alternatively one can use the following characterization: ∀ε > 0 ∈ K: ∃n ∈ N:1/n < ε. The qualifier "Archimedean" is formulated in the theory of rank one valued fields and normed spaces over rank one valued fields as follows. Let F be a field endowed with an absolute value function, i.e. a function which associates the real number 0 with the field element 0 and associates a positive real number | x | with each non-zero x ∈ F and satisfies | x y | = | x | | y | and | x + y | ≤ | x | + | y |.
F is said to be Archimedean if for any non-zero x ∈ F there exists a natural number n such that | x + ⋯ + x ⏟ n terms | > 1. A normed space is Archimedean if a sum of n terms, each equal to a non-zero vector x, has norm greater than one for sufficiently large n. A field with an absolute value or a normed space is either Archimedean or satisfies the stronger condition, referred to as the ultrametric triangle inequality, | x + y | ≤ max,respectively. A field or normed space satisfying the ultrametric triangle inequality is called non-Archimedean; the concept of a non-Archimedean normed linear space was introduced by A. F. Monna; the field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function | x |