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
Marshall Hall (mathematician)
Marshall Hall, Jr. was an American mathematician who made significant contributions to group theory and combinatorics. He studied mathematics at Yale University, graduating in 1932, he studied for a year at Cambridge University under a Henry Fellowship working with G. H. Hardy, he returned to Yale to take his Ph. D. in 1936 under the supervision of Øystein Ore. He worked in Naval Intelligence during World War II, including six months in 1944 at Bletchley Park, the center of British wartime code breaking. In 1946 he took a position at The Ohio State University. In 1959 he moved to the California Institute of Technology where, in 1973, he was named the first IBM Professor at Caltech, the first named chair in mathematics. After retiring from Caltech in 1981, he accepted a post at Emory University in 1985. Hall died in 1990 in London on his way to a conference to mark his 80th birthday, he wrote a number of papers of fundamental importance in group theory, including his solution of Burnside's problem for groups of exponent 6, showing that a finitely generated group in which the order of every element divides 6 must be finite.
His work in combinatorics includes an important paper of 1943 on projective planes, which for many years was one of the most cited mathematics research papers. In this paper he constructed a family of non-Desarguesian planes which are known today as Hall planes, he worked on block designs and coding theory. His classic book on group theory was well received when it is still useful today, his book Combinatorial Theory came out in a second edition in 1986, published by John Sons. He proposed Hall's conjecture on the differences between perfect squares and perfect cubes, which remains an open problem as of 2015. 1943: "Projective Planes", Transactions of the American Mathematical Society 54: 229–77 doi:10.2307/1990331 1959: The Theory of Groups, Macmillan MR103215 Wilhelm Magnus Review: Marshall Hall, Jr. Theory of Groups Bulletin of the American Mathematical Society 66: 144–6. 1964: The Groups of Order 2n n ≤ 6), Macmillan MR168631 Preface: "An exhaustive catalog of the 340 groups of order dividing 64 with detailed tables of defining relations and lattice presentations of each group in the notation the text defines.
"Of enduring value to those interested in finite groups". 1967: Combinatorial Theory, Blaisdell MR224481 Hall, Jr. Marshall, "Mathematical Biography: Marshall Hall Jr.", in Duran, Peter. A Century of mathematics in America, vol 1, Providence, RI: American Mathematical Society, pp. 367–374, ISBN 0-8218-0124-4 Zassenhaus, Hans, "Marshall Hall, Jr.: 1910–1990", Notices of the American Mathematical Society, 37: 1033, ISSN 0002-9920, MR 1071446 O'Connor, John J.. Marshall Hall at the Mathematics Genealogy Project
In classical cryptography, the Hill cipher is a polygraphic substitution cipher based on linear algebra. Invented by Lester S. Hill in 1929, it was the first polygraphic cipher in which it was practical to operate on more than three symbols at once; the following discussion assumes an elementary knowledge of matrices. Each letter is represented by a number modulo 26; the simple scheme A = 0, B = 1... Z = 25 is used. To encrypt a message, each block of n letters is multiplied by an invertible n × n matrix, against modulus 26. To decrypt the message, each block is multiplied by the inverse of the matrix used for encryption; the matrix used for encryption is the cipher key, it should be chosen randomly from the set of invertible n × n matrices. The cipher can, of course, be adapted to an alphabet with any number of letters. Consider the message'ACT', the key below: Since'A' is 0,'C' is 2 and'T' is 19, the message is the vector: Thus the enciphered vector is given by: = ≡ which corresponds to a ciphertext of'POH'.
Now, suppose that our message is instead'CAT', or: This time, the enciphered vector is given by: ≡ ≡ which corresponds to a ciphertext of'FIN'. Every letter has changed; the Hill cipher has achieved Shannon's diffusion, an n-dimensional Hill cipher can diffuse across n symbols at once. In order to decrypt, we turn the ciphertext back into a vector simply multiply by the inverse matrix of the key matrix. We find that, modulo 26, the inverse of the matrix used in the previous example is: − 1 ≡ Taking the previous example ciphertext of'POH', we get: ≡ ≡ which gets us back to'ACT', just as we hoped. We have not yet discussed two complications. Not all matrices have an inverse; the matrix will have an inverse if and only. In the case of the Hill Cipher, the determinant of the encrypting matrix must not have any common factors with the modular base. Thus, if w
Hans Julius Zassenhaus was a German mathematician, known for work in many parts of abstract algebra, as a pioneer of computer algebra. He was born in Koblenz in 1912, his father was a advocate for Reverence for Life as expressed by Albert Schweitzer. Hans had two brothers and Wilfred, sister Hiltgunt, who wrote an autobiography in 1974. According to her, their father lost his position as school principal due to his philosophy, she wrote: Hans, my eldest brother, studied mathematics. My brothers Guenther and Wilfred were in medical school.... Only students who participated in Nazi activities would get scholarships; that left us out. Together we made an all-out effort.... Soon our house became a beehive. Day in and day out for the next four years a small army of children of all ages would arrive to be tutored. At the University of Hamburg Zassenhaus came under the influence of Emil Artin; as he wrote later: His introductory course in analysis that I attended at the age of 17 converted me from a theoretical physicist to a mathematician.
When just 21, Zassenhaus was studying composition series in group theory. He proved his butterfly lemma that provides a refinement of two normal chains to isomorphic central chains. Inspired by Artin, Zassenhaus wrote a textbook Lehrbuch der Gruppentheorie, translated as Theory of Groups, his thesis was on doubly transitive permutation groups with Frobenius groups as stabilizers. These groups are now called Zassenhaus groups, they have had a deep impact on the classification of finite simple groups. He obtained his doctorate in June 1934, took the teachers’ exam the next May, he became a scientific assistant at University of Rostock. In 1936 he became assistant to Artin back in Hamburg, but Artin departed for the USA the following year. Zassenhaus gave his Habilitation in 1938. According to his sister Hiltgunt, Hans was "called up as a research scientist at a weather station" for his part in the German war effort. Zassenhaus married Lieselotte Lohmann in 1942; the couple raised three children: Michael and Peter.
In 1943 Zassenhaus became extraordinary professor. He became Managing Director of the Hamburg Mathematical Seminar. After the war, as a fellow of the British Council, Zassenhaus visited University of Glasgow in 1948. There he was given an honorary Master of Arts degree; the following year he joined the faculty of McGill University where the endowments of Peter Redpath financed a professorship. He was at McGill for a decade with leaves of absence to Institute for Advanced Study and California Institute of Technology. There he was using computers to advance number theory. In 1959 Zassenhaus began teaching at University of Notre Dame and became director of its computing center in 1964. Zassenhaus was a Mershon visiting professor at Ohio State University in the fall of 1963. In 1965 he came to Ohio State permanently; the mathematics department was led by Arnold Ross. Nonetheless, he continued to take leaves of absence for visits to Göttingen, Heidelberg, UCLA, Warwick, CIT, U Montreal, Saarbrücken, he served as editor in chief of the Journal of Number Theory from its first issue in 1967.
He won a Lester R. Ford Award in 1968. Hans Zassenhaus died in Columbus, Ohio on November 21, 1991, his doctoral students include Joachim Lambek. Hans Julius Zassenhaus, Lehrbuch der Gruppentheorie, 2nd edition,The theory of groups. A famous group theory book based on a course by Emil Artin given at the University of Hamburg during winter semester 1933 and summer semester 1934. Zassenhaus showed that there are just seven near-fields that are not division rings or Dickson near-fields in Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 11, pp 187–220. In 1977 Academic Press published Number Theory and Algebra, a collection of papers dedicated to Henry B. Mann, Arnold E. Ross, Olga Taussky-Todd, edited by Zassenhaus, it included "A Theorem on Cyclic Algebras" by Zassenhaus. Cambridge University Press published Algorithmic Algebraic Number Theory written by Zassenhaus and M. Pohst in 1989. A second edition appeared in 1993. Pohst, M.. 1st paperback edition. ISBN 978-0-521-59669-5. Cantor, David G..
The paper that introduced the Cantor–Zassenhaus algorithm for factoring polynomials. Zassenhaus dual expansion Schur–Zassenhaus theorem M. Pohst "Hans Zassenhaus", Journal of Number Theory 47:1–19. O'Connor, John J.. "Hans Zassenhaus", MacTutor History of Mathematics archive, University of St Andrews. Biography from the Ohio State University
In three-dimensional space, a Platonic solid is a regular, convex polyhedron. It is constructed by congruent regular polygonal faces with the same number of faces meeting at each vertex. Five solids meet these criteria: Geometers have studied the Platonic solids for thousands of years, they are named for the ancient Greek philosopher Plato who hypothesized in his dialogue, the Timaeus, that the classical elements were made of these regular solids. The Platonic solids have been known since antiquity, it has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes. The ancient Greeks studied the Platonic solids extensively; some sources credit Pythagoras with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.
The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote about them in the dialogue Timaeus c.360 B. C. in which he associated each of the four classical elements with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing. Air is made of the octahedron. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a nonspherical solid, the hexahedron represents "earth"; these clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube's being the only regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven".
Aristotle added a fifth element, aithēr and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid. Euclid mathematically described the Platonic solids in the Elements, the last book of, devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, cube and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues. Andreas Speiser has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in the Elements. Much of the information in Book XIII is derived from the work of Theaetetus. In the 16th century, the German astronomer Johannes Kepler attempted to relate the five extraterrestrial planets known at that time to the five Platonic solids. In Mysterium Cosmographicum, published in 1596, Kepler proposed a model of the Solar System in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres.
Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids enclosed within a sphere that represented the orbit of Saturn. The six spheres each corresponded to one of the planets; the solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron and the cube, thereby dictating the structure of the solar system and the distance relationships between the planets by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of, that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy, he discovered the Kepler solids. In the 20th century, attempts to link Platonic solids to the physical world were expanded to the electron shell model in chemistry by Robert Moon in a theory known as the "Moon model". For Platonic solids centered at the origin, simple Cartesian coordinates of the vertices are given below.
The Greek letter φ is used to represent the golden ratio 1 + √5/2 ≈ 1.6180. The coordinates for the tetrahedron and dodecahedron are given in two orientation sets, each containing half of the sign and position permutation of coordinates; these coordinates reveal certain relationships between the Platonic solids: the vertices of the tetrahedron represent half of those of the cube, as or, one of two sets of 4 vertices in dual positions, as h or. Both tetrahedral positions make the compound stellated octahedron; the coordinates of the icosahedron are related to two alternated sets of coordinates of a nonuniform truncated octahedron, t or called a snub octahedron, as s or, seen in the compound of two icosahedra. Eight of the vertices of the dodecahedron are shared with the cube. Completing all orientat
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
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the world's oldest publishing house and the second-largest university press in the world, it holds letters patent as the Queen's Printer. The press mission is "to further the University's mission by disseminating knowledge in the pursuit of education and research at the highest international levels of excellence". Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global sales presence, publishing hubs, offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries, its publishing includes academic journals, reference works and English language teaching and learning publications. Cambridge University Press is a charitable enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press.
It originated from letters patent granted to the University of Cambridge by Henry VIII in 1534, has been producing books continuously since the first University Press book was printed. Cambridge is one of the two privileged presses. Authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, Stephen Hawking. University printing began in Cambridge when the first practising University Printer, Thomas Thomas, set up a printing house on the site of what became the Senate House lawn – a few yards from where the press's bookshop now stands. In those days, the Stationers' Company in London jealously guarded its monopoly of printing, which explains the delay between the date of the university's letters patent and the printing of the first book. In 1591, Thomas's successor, John Legate, printed the first Cambridge Bible, an octavo edition of the popular Geneva Bible; the London Stationers objected strenuously. The university's response was to point out the provision in its charter to print "all manner of books".
Thus began the press's tradition of publishing the Bible, a tradition that has endured for over four centuries, beginning with the Geneva Bible, continuing with the Authorized Version, the Revised Version, the New English Bible and the Revised English Bible. The restrictions and compromises forced upon Cambridge by the dispute with the London Stationers did not come to an end until the scholar Richard Bentley was given the power to set up a'new-style press' in 1696. In July 1697 the Duke of Somerset made a loan of £200 to the university "towards the printing house and presse" and James Halman, Registrary of the University, lent £100 for the same purpose, it was in Bentley's time, in 1698, that a body of senior scholars was appointed to be responsible to the university for the press's affairs. The Press Syndicate's publishing committee still meets and its role still includes the review and approval of the press's planned output. John Baskerville became University Printer in the mid-eighteenth century.
Baskerville's concern was the production of the finest possible books using his own type-design and printing techniques. Baskerville wrote, "The importance of the work demands all my attention. Caxton would have found nothing to surprise him if he had walked into the press's printing house in the eighteenth century: all the type was still being set by hand. A technological breakthrough was badly needed, it came when Lord Stanhope perfected the making of stereotype plates; this involved making a mould of the whole surface of a page of type and casting plates from that mould. The press was the first to use this technique, in 1805 produced the technically successful and much-reprinted Cambridge Stereotype Bible. By the 1850s the press was using steam-powered machine presses, employing two to three hundred people, occupying several buildings in the Silver Street and Mill Lane area, including the one that the press still occupies, the Pitt Building, built for the press and in honour of William Pitt the Younger.
Under the stewardship of C. J. Clay, University Printer from 1854 to 1882, the press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks. During Clay's administration, the press undertook a sizeable co-publishing venture with Oxford: the Revised Version of the Bible, begun in 1870 and completed in 1885, it was in this period as well that the Syndics of the press turned down what became the Oxford English Dictionary—a proposal for, brought to Cambridge by James Murray before he turned to Oxford. The appointment of R. T. Wright as Secretary of the Press Syndicate in 1892 marked the beginning of the press's development as a modern publishing business with a defined editorial policy and administrative structure, it was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories. The Cambridge Modern History was published