# Combinatorics

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Cayley's theorem Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) as a permutation group of some underlying set. Thus, theorems that are true for subgroups of permutation groups are true for groups in general. Nevertheless, Alperin and Bell note that "in general the fact that finite groups are imbedded in symmetric groups has not influenced the methods used to study finite groups".[4] The regular action used in the standard proof of Cayley's theorem does not produce the representation of G in a minimal-order permutation group. , itself already a symmetric group of order 6, would be represented by the regular action as a subgroup of (a group of order 720).[5] The problem of finding an embedding of a group in a minimal-order symmetric group is rather more difficult.[6][7] History The theorem was later published by Walther Dyck in 1882[12] and is attributed to Dyck in the first edition of Burnside's book.[13] . Firstly, suppose with .

Statistical physics Statistical physics is a branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Its applications include many problems in the fields of physics, biology, chemistry, neurology, and even some social sciences, such as sociology. Statistical mechanics Statistical mechanics provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties of materials that can be observed in everyday life, therefore explaining thermodynamics as a natural result of statistics, classical mechanics, and quantum mechanics at the microscopic level. One of the most important equations in statistical mechanics (analogous to , which is essentially a weighted sum of all possible states available to a system. where . See also

Probability theory Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data.[1] Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. History of probability Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. . . . If is

Symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols.[1] Since there are is Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics. is isomorphic to a subgroup of the symmetric group on (the underlying set of) Definition and first properties The symmetric group on a finite set is the group whose elements are all bijective functions from to is the symmetric group on the set The symmetric group on a set is denoted in various ways, including and .[1] If is the set , or ). .

Probability interpretations The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical tendency of something to occur or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory. There are two broad categories[1][2] of probability interpretations which can be called "physical" and "evidential" probabilities. Physical probabilities, which are also called objective or frequency probabilities, are associated with random physical systems such as roulette wheels, rolling dice and radioactive atoms. In such systems, a given type of event (such as a die yielding a six) tends to occur at a persistent rate, or "relative frequency", in a long run of trials. It is unanimously agreed that statistics depends somehow on probability. Philosophy [2] (p 1132) If we denote by

Bijection Function that is one to one and onto (mathematics) A bijective function, f: X → Y, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, f(1) = D. An injective surjective function (bijection) A non-injective surjective function (surjection, not a bijection) A non-injective non-surjective function (also not a bijection) A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements. A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms a symmetry group. Bijective functions are essential to many areas of mathematics including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group, and projective map. Definition For a pairing between X and Y (where Y need not be different from X) to be a bijection, four properties must hold: Examples Inverses is Notes

Game of chance Roulette is a game of chance, no strategy can give players advantages, the outcome is determined by pure chance A game of chance is a game whose outcome is strongly influenced by some randomizing device, and upon which contestants may choose to wager money or anything of monetary value. Common devices used include dice, spinning tops, playing cards, roulette wheels, or numbered balls drawn from a container. A game of chance may have some skill element to it, however, chance generally plays a greater role in determining the outcome than skill. A game of skill, on the other hand, also may have elements of chance, but with skill playing a greater role in determining the outcome. Any game of chance that involves anything of monetary value is gambling. Gambling is known in nearly all human societies, even though many have passed laws restricting it. Addiction He must play regularly: the issue here is to know from when the subject performs "too much." See also References

Group A Group is a number of people or things that are located, gathered, or classed together. Groups of people Social group, an entity of two or more people that strives through collaborative empiricismEthnic group, an entity with members sharing similar phenotype, lingua, lineage and sociocultural experiencesOrganization, an entity that has a collective goal and is linked to an external environment Groups of animals Science and technology Mathematics Chemistry Computing and the Internet Other uses in science and technology Other uses See also