In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group, generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, it contains an element g such that every other element of the group may be obtained by applying the group operation to g or its inverse; each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group; every infinite cyclic group is isomorphic to the additive group of the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n; every cyclic group is an abelian group, every finitely generated abelian group is a direct product of cyclic groups. Every cyclic group of prime order is a simple group. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order; the cyclic groups of prime order are thus among the building blocks from which all groups can be built.
For any element g in any group G, one can form the subgroup of all integer powers ⟨g⟩ =, called the cyclic subgroup of g. The order of g is the number of elements in ⟨g⟩. A cyclic group is a group, equal to one of its cyclic subgroups: G = ⟨g⟩ for some element g, called a generator. For a finite cyclic group with order |G| = n, this means G =, where e is the identity element and gj = gk whenever j ≡ k modulo n. An abstract group defined by this multiplication is denoted Cn, we say that G is isomorphic to the standard cyclic group Cn; such a group is isomorphic to Z/nZ, the group of integers modulo n with the addition operation, the standard cyclic group in additive notation. Under the isomorphism χ defined by χ = i the identity element e corresponds to 0, products correspond to sums, powers correspond to multiples. For example, the set of complex 6th roots of unity G = forms a group under multiplication, it is cyclic, since it is generated by the primitive root z = 1 2 + 3 2 i = e 2 π i / 6: that is, G = ⟨z⟩ = with z6 = 1.
Under a change of letters, this is isomorphic to the standard cyclic group of order 6, defined as C6 = ⟨g⟩ = with multiplication gj · gk = gj+k, so that g6 = g0 = e. These groups are isomorphic to Z/6Z = with the operation of addition modulo 6, with zk and gk corresponding to k. For example, 1 + 2 ≡ 3 corresponds to z1 · z2 = z3, 2 + 5 ≡ 1 corresponds to z2 · z5 = z7 = z1, so on. Any element generates its own cyclic subgroup, such as ⟨z2⟩ = of order 3, isomorphic to C3 and Z/3Z. Instead of the quotient notations Z/nZ, Z/, or Z/n, some authors denote a finite cyclic group as Zn, but this conflicts with the notation of number theory, where Zp denotes a p-adic number ring, or localization at a prime ideal. On the other hand, in an infinite cyclic group G = ⟨g⟩, the powers gk give distinct elements for all integers k, so that G =, G is isomorphic to the standard group C = C∞ and to Z, the additive group of the integers. An example is the first frieze group. Here there are no finite cycles, the name "cyclic" may be misleading.
To avoid this confusion, Bourbaki introduced the term monogenous group for a group with a single generator and restricted "cyclic group" to mean a finite monogenous group, avoiding the term "infinite cyclic group". The set of integers Z,with the operation of addition, forms a group, it is an infinite cyclic group, because all integers can be written by adding or subtracting the single number 1. In this group, 1 and −1 are the only generators; every infinite cyclic group is isomorphic to Z. For every positive integer n, the set of integers modulo n, again with the operation of addition, forms a finite cyclic group, denoted Z/nZ. A modular integer i is a generator of this group if i is prime to n, because these elements can generate all other elements of the group through integer addition; every finite cyclic group G is isomorphic to Z/nZ. The addition operations on integers and modular integers, used to define the cyclic groups, are the addition operations of commutative rings denoted Z and Z/nZ or Z/.
If p is a prime Z/pZ is a finite field, is denoted Fp or GF. For every positive integer n, the set of the integers modulo n that are prime to n is written as ×; this group is not always cyclic, bu
In the mathematical field of graph theory, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f: V → V such that f = v 2. In other words, a graph is vertex-transitive if its automorphism group acts transitively upon its vertices. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical; every symmetric graph without isolated vertices is vertex-transitive, every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric, not all regular graphs are vertex-transitive. Finite vertex-transitive graphs include the symmetric graphs; the finite Cayley graphs are vertex-transitive, as are the vertices and edges of the Archimedean solids. Potočnik and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices. Although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs.
The most famous example is the Petersen graph, but others can be constructed including the line graphs of edge-transitive non-bipartite graphs with odd vertex degrees. The edge-connectivity of a vertex-transitive graph is equal to the degree d, while the vertex-connectivity will be at least 2/3. If the degree is 4 or less, or the graph is edge-transitive, or the graph is a minimal Cayley graph the vertex-connectivity will be equal to d. Infinite vertex-transitive graphs include: infinite paths infinite regular trees, e.g. the Cayley graph of the free group graphs of uniform tessellations, including all tilings by regular polygons infinite Cayley graphs the Rado graphTwo countable vertex-transitive graphs are called quasi-isometric if the ratio of their distance functions is bounded from below and from above. A well known conjecture stated that every infinite vertex-transitive graph is quasi-isometric to a Cayley graph. A counterexample was proposed by Diestel and Leader in 2001. In 2005, Eskin and Whyte confirmed the counterexample.
Edge-transitive graph Lovász conjecture Semi-symmetric graph Zero-symmetric graph Weisstein, Eric W. "Vertex-transitive graph". MathWorld. A census of small connected cubic vertex-transitive graphs. Primož Potočnik, Pablo Spiga, Gabriel Verret, 2012
In graph theory, a branch of mathematics, an undirected graph is called an asymmetric graph if it has no nontrivial symmetries. Formally, an automorphism of a graph is a permutation p of its vertices with the property that any two vertices u and v are adjacent if and only if p and p are adjacent; the identity mapping of a graph onto itself is always an automorphism, is called the trivial automorphism of the graph. An asymmetric graph is a graph; the smallest asymmetric non-trivial graphs have 6 vertices. The smallest asymmetric regular graphs have ten vertices. One of the two smallest asymmetric cubic graphs is the twelve-vertex Frucht graph discovered in 1939. According to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs; the class of asymmetric graphs is closed under complements: a graph G is asymmetric if and only if its complement is. Any n-vertex asymmetric graph can be made symmetric by adding and removing a total of at most n/2 + o edges; the proportion of graphs on n vertices with nontrivial automorphism tends to zero as n grows, informally expressed as "almost all finite graphs are asymmetric".
In contrast, again informally, "almost all infinite graphs are symmetric." More countable infinite random graphs in the Erdős–Rényi model are, with probability 1, isomorphic to the symmetric Rado graph. The smallest asymmetric tree has seven vertices: it consists of three paths of lengths 1, 2, 3, linked at a common endpoint. In contrast to the situation for graphs all trees are symmetric. In particular, if a tree is chosen uniformly at random among all trees on n labeled nodes with probability tending to 1 as n increases, the tree will contain some two leaves adjacent to the same node and will have symmetries exchanging these two leaves
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form under the operation of matrix multiplication. Elements a, b and c can be taken from any commutative ring with identity taken to be the ring of real numbers or the ring of integers; the continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems in the context of the Stone–von Neumann theorem. More one can consider Heisenberg groups associated to n-dimensional systems, most to any symplectic vector space. In the three-dimensional case, the product of two Heisenberg matrices is given by: =. Since the multiplication is not commutative, the group is non-abelian; the neutral element of the Heisenberg group is the identity matrix, inverses are given by − 1 =. The group is a subgroup of the 2-dimensional affine group Aff: acting on corresponds to the affine transform x → +. There are several prominent examples of the three-dimensional case.
If a, b, c, are real numbers one has the continuous Heisenberg group H3. It is a nilpotent real Lie group of dimension 3. In addition to the representation as real 3x3 matrices, the continuous Heisenberg group has several different representations in terms of function spaces. By Stone–von Neumann theorem, there is, up to isomorphism, a unique irreducible unitary representation of H in which its centre acts by a given nontrivial character; this representation has models. In the Schrödinger model, the Heisenberg group acts on the space of square integrable functions. In the theta representation, it acts on the space of holomorphic functions on the upper half-plane. If a, b, c, are integers one has the discrete Heisenberg group H3, it is a non-abelian nilpotent group. It has two generators, x =, y = and relations z = x y x − 1 y − 1, x z = z x, y z = z y,where z =
In graph-theoretic mathematics, a biregular graph or semiregular bipartite graph is a bipartite graph G = for which every two vertices on the same side of the given bipartition have the same degree as each other. If the degree of the vertices in U is x and the degree of the vertices in V is y the graph is said to be -biregular; every complete bipartite graph K a, b is -biregular. The rhombic dodecahedron is another example. An -biregular graph G = must satisfy the equation x | U | = y | V |; this follows from a simple double counting argument: the number of endpoints of edges in U is x | U |, the number of endpoints of edges in V is y | V |, each edge contributes the same amount to both numbers. Every regular bipartite graph is biregular; every edge-transitive graph, not vertex-transitive must be biregular. In particular every edge-transitive graph is either biregular; the Levi graphs of geometric configurations are biregular.
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Arthur Cayley was a British mathematician. He helped; as a child, Cayley enjoyed solving complex maths problems for amusement. He entered Trinity College, where he excelled in Greek, French and Italian, as well as mathematics, he worked as a lawyer for 14 years. He postulated the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, verified it for matrices of order 2 and 3, he was the first to define the concept of a group in the modern way—as a set with a binary operation satisfying certain laws. When mathematicians spoke of "groups", they had meant permutation groups. Cayley tables and Cayley graphs as well. Arthur Cayley was born in Richmond, England, on 16 August 1821, his father, Henry Cayley, was a distant cousin of Sir George Cayley, the aeronautics engineer innovator, descended from an ancient Yorkshire family. He settled in Russia, as a merchant, his mother was daughter of William Doughty. According to some writers she was Russian, his brother was the linguist Charles Bagot Cayley.
Arthur spent his first eight years in Saint Petersburg. In 1829 his parents were settled permanently near London. Arthur was sent to a private school. At age 14 he was sent to King's College School; the school's master observed indications of mathematical genius and advised the father to educate his son not for his own business, as he had intended, but to enter the University of Cambridge. At the unusually early age of 17 Cayley began residence at Cambridge; the cause of the Analytical Society had now triumphed, the Cambridge Mathematical Journal had been instituted by Gregory and Robert Leslie Ellis. To this journal, at the age of twenty, Cayley contributed three papers, on subjects, suggested by reading the Mécanique analytique of Lagrange and some of the works of Laplace. Cayley's tutor at Cambridge was George Peacock and his private coach was William Hopkins, he finished his undergraduate course by winning the place of Senior Wrangler, the first Smith's prize. His next step was to take the M.
A. degree, win a Fellowship by competitive examination. He continued to reside at Cambridge University for four years; because of the limited tenure of his fellowship it was necessary to choose a profession. He made a specialty of conveyancing, it was while he was a pupil at the bar examination that he went to Dublin to hear Hamilton's lectures on quaternions. His friend J. J. Sylvester, his senior by five years at Cambridge, was an actuary, resident in London. During this period of his life, extending over fourteen years, Cayley produced between two and three hundred papers. At Cambridge University the ancient professorship of pure mathematics is denominated by the Lucasian, is the chair, occupied by Isaac Newton. Around 1860, certain funds bequeathed by Lady Sadleir to the University, having become useless for their original purpose, were employed to establish another professorship of pure mathematics, called the Sadleirian; the duties of the new professor were defined to be "to explain and teach the principles of pure mathematics and to apply himself to the advancement of that science."
To this chair Cayley was elected. He gave up a lucrative practice for a modest salary, he at once settled down in Cambridge. More fortunate than Hamilton in his choice, his home life was one of great happiness, his friend and fellow investigator, once remarked that Cayley had been much more fortunate than himself. At first the teaching duty of the Sadleirian professorship was limited to a course of lectures extending over one of the terms of the academic year. For many years the attendance was small, came entirely from those who had finished their career of preparation for competitive examinations; the subject lectured on was that of the memoir on which the professor was for the time engaged. The other duty of the chair — the advancement of mathematical science — was discharged in a handsome manner by the long series of memoirs that he published, ranging over every department of pure mathematics, but it was discharged in a much less obtrusive way. In 1872 he was made an honorary fellow of Trinity College, three years an ordinary fellow, which meant stipend as well as honour.
About this time his friends subscribed for a presentation portrait. Maxwell wrote an address to the committee of subscribers; the verses refer to the subjects investigated in several of Cay