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# Topic: Classification of finite simple groups

###### In the News (Thu 27 Apr 17)

 Encyclopedia: Classification of finite simple groups In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. In mathematics, the unitary group of degree n over the field F (which is either the field R of real numbers or the field C of complex numbers) is the group of n by n unitary matrices with entries from F, with the group operation that of matrix multiplication. Finite groups The American Mathematical Society (AMS) is dedicated to the interests of mathematical research and education, which it does with various publications and conferences as well as annual monetary awards to mathematicians. www.nationmaster.com /encyclopedia/Classification-of-finite-simple-groups   (2066 words)

 Finite group - Wikipedia, the free encyclopedia Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable groups and nilpotent groups. Finite groups are directly relevant to symmetry, when that is restricted to a finite number of transformations. In this way, finite groups and their properties can enter centrally in questions, for example in theoretical physics, where their role is not initially obvious. en.wikipedia.org /wiki/Finite_group   (243 words)

 Classification of finite simple groups - Wikipedia, the free encyclopedia The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. Gorenstein announced the classification of finite simple groups in 1983, based partly on the impression that the quasithin case was finished. In contrast, during the original proof, nobody knew how many sporadic groups there were, and in fact some of the sporadic groups (for example, the Janko groups) were discovered in the process of trying to prove cases of the classification theorem. en.wikipedia.org /wiki/Classification_of_finite_simple_groups   (893 words)

 PlanetMath: examples of finite simple groups The first trivial example of simple groups are the cyclic groups of prime order. The simplicity of the alternating groups is an important fact that Évariste Galois required in order to prove the insolubility by radicals of the general polynomial of degree higher than four. This is version 12 of examples of finite simple groups, born on 2002-11-04, modified 2004-11-17. www.planetmath.org /encyclopedia/ExamplesOfFiniteSimpleGroups.html   (360 words)

 Classification of finite simple groups   (Site not responding. Last check: 2007-10-20) The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which classifies all of the finite simple group s. One of 26 left-over groups known as the sporadic groups The theorem has widespread applications in many branches of applications, as questions about finite groups can often be reduced to questions about finite simple groups, which by the classification can be reduced to an enumeration of cases. Finite Simple Groups "The classification of the finite simple groups" by D. Gorenstein, R. Lyons, and R. Solomon in PDF. www.serebella.com /encyclopedia/article-Classification_of_finite_simple_groups.html   (805 words)

 Classification of finite simple groups - Open Encyclopedia   (Site not responding. Last check: 2007-10-20) The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which classifies all of the finite simple groups. The theorem has widespread applications in many branches of applications, as questions about finite groups can often be reduced to questions about finite simple groups, which by the classification can be reduced to an enumeration of cases. In particular, during the original proof, nobody knew how many sporadic groups there would be, and in fact some of the sporadic groups (for example, the Janko groups) were discovered in the process of trying to prove the classification theorem. www.open-encyclopedia.com /Classification_of_finite_simple_groups   (432 words)

 Classification of finite simple groups by Aschbacher The Status of the Classification of Finite Simple Groups: Finite simple groups have a special significance in mathematics as the "building blocks" for all finite groups, in much the same way that the prime numbers are the "building blocks" for the integers. The problem of classifying all finite simple groups comes down to proving that they fall into four distinct classes: groups of prime order, alternating groups, groups of Lie type, and sporadic groups (of which there are 26). www.ams.org /ams/aschbacher.html   (295 words)

 Classification of finite simple groups - Encyclopedia.WorldSearch   (Site not responding. Last check: 2007-10-20) sporadic groups there would be, and in fact some of the sporadic groups (for example, the Janko groups) were discovered in the process of trying to prove the classification theorem. Michael Aschbacher, The Status of the Classificatin of the Finite Simple Groups, Notices of the American Mathematical Society, August 2004 The Classification Of Quasithin Groups: II: The Classification of Simple QTKE-groups (Mathematical Surveys and Monographs) encyclopedia.worldsearch.com /classification_of_finite_simple_groups.htm   (628 words)

 Classification of finite simple groups   (Site not responding. Last check: 2007-10-20) The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which classifies all of the finite simplegroups. Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21were found between 1965 and 1975. In particular, during the original proof, nobody knew how many sporadic groups there would be, and in factsome of the sporadic groups (for example, the Janko groups) were discoveredin the process of trying to prove the classification theorem. www.therfcc.org /classification-of-finite-simple-groups-78127.html   (414 words)

 Classification of finite simple groups   (Site not responding. Last check: 2007-10-20) The theorem has widespread applications in many of applications as questions about finite groups often be reduced to questions about finite groups which by the classification can be to an enumeration of cases. Five of the sporadic groups were discovered Mathieu in the 1860s and the other 21 were found 1965 and 1975. particular during the original proof nobody knew many sporadic groups there would be and fact some of the sporadic groups (for the Janko groups) were discovered in the process of to prove the classification theorem. www.freeglossary.com /Classification_of_finite_simple_groups   (609 words)

 The Classification of Finite Simple Groups, Vol 1 New, Used Books, Cheap Prices, ISBN 0306413051 The Classification of the Finite Simple Groups, Nu... Classification of the Finite Simple Groups (By Daniel Gorenstein) The Classification of the Finite Simple Groups, Vo... www.bookfinder4u.com /detail/0306413051.html   (228 words)

 The Ultimate Simple group - American History Information Guide and Reference   (Site not responding. Last check: 2007-10-20) The only abelian simple groups are the cyclic groups of prime order. Therefore every finite simple group has even order unless it is cyclic of prime order. Simple groups of infinite order also exist: simple Lie groups and the infinite Thompson groups T and V are examples of these. www.historymania.com /american_history/Simple_group   (176 words)

 Finite simple groups   (Site not responding. Last check: 2007-10-20) The finite simple groups, namely those which have no nontrivial normal subgroups, are the building blocks for all finite groups. Thus it is important to have a deep understanding of properties of the finite simple groups. The bulk of the finite simple groups are the groups of Lie type, and for these groups there are fruitful ways of applying the work on algebraic groups outlined above. www.ma.ic.ac.uk /~mwl/finite.htm   (143 words)

 AMCA: Finite Simple Groups, Geometries, Buildings, and Related Topics, Conference in Honor of Ernest Shult - All ... Let E(g) be the set of all simple groups S such that S is a composition factor of a group of genus g and S is neither cyclic nor alternating. The majority of simple groups in E(0) are of low Lie rank, and in the majority of cases the related action is on points of the natural module. In 1992, the symmetric genus of 23 of the sporadic simple groups was determined. www.math.ksu.edu /conf2001/abstract.html   (3711 words)

 20D: Abstract Finite Groups Moreover, we study the internal properties of those groups -- material on their representation (20C) or permutation actions (20B) or cohomology (20J) are for now on the main group theory page. Finite groups with distinct but isomorphic characteristic subgroups All finite simple groups can be generated by two elements; indeed for alternating (and symmetric) groups, such pairs of generators are legion. www.math.niu.edu /~rusin/known-math/index/20DXX.html   (707 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-10-20) The classification of the finite simple groups has been basically what has driven the study of abstract algebra in the 20th century. The finite simple groups are: Z_p where p is prime - the integers modulo a prime. Some more information can be gotten on the Monster, and finite simple groups in general, by reading this book: Mathematical Surveys and Monographs: The Classification of the Finite Simple Groups by Daniel Gorenstein, Lyons and Solomon, The American Mathematical Society, 1994. mathforum.org /library/drmath/view/51459.html   (279 words)

 Group Theory & Rubik's Cube Group theory is the study of the algebra of transformations and symmetry. Given an element x of a group G, the orbit of x is the set of all elements of G which are generated by x, i.e. A representation of a group G is a set of matrices M which are homomorphic to the group. akbar.marlboro.edu /~mahoney/courses/Spr00/rubik.html   (3602 words)

 Citations: The structure of models of uncountably categorical theories - Zil'ber (ResearchIndex)   (Site not responding. Last check: 2007-10-20) Base transitive groups of rank 1 are just regular permutation groups (possibly with some global fixed points) Theorem 7.1 For a base transitive group G, the p. Her proof uses the classification of finite simple groups to determine the rank 2 geometric groups, and then proceeds by induction. Maund s results have not been published (except in her thesis) but a survey of the methods and results in the rank 2 case is given by Cameron [1] The relevance of geometric groups to our problem is given by the next result, whose proof is immediate from the properties of independent sets. citeseer.ist.psu.edu /context/1034025/0   (484 words)

 [No title]   (Site not responding. Last check: 2007-10-20) Title: The Classification of Finite Simple Groups: Aspects of the Second Generation Proof Abstract. The classification of finite simple groups is widely acknowledged to be one of the major results in modern mathematics. In this talk we will outline the "Generation 2"-proof of the Classification, and discuss a specific part of it, in which the speaker is involved. www.math.hmc.edu /~jacobsen/ik.txt   (135 words)

 Amazon.ca: Books: The Classification of the Finite Simple Groups: Part Ii, Chapter G : General Group Therapy   (Site not responding. Last check: 2007-10-20) The Classification Theorem is one of the main achievements of 20th century mathematics, but its proof has not yet been completely extricated from the journal literature in which it first appeared. This is the second volume in a series devoted to the presentation of a reorganized and simplified proof of the classification of the finite simple groups. The authors present (with either proof or reference to a proof) those theorems of abstract finite group theory, which are fundamental to the analysis in later volumes in the series. pdxbooks.com /send/s27/0821803905   (370 words)

 groups Examples of Finite Groups: A = {1, -1, i, -i} where * is multiplication, B = {0, 1, 2, 3) where * is addition modulo 4. A subgroup is a group entirely inside another: {1, -1} is a subgroup of A, {0, 2) is is a subgroup of B. The makers of GAP have written an analysis of Rubik's Cube from a Group Theory perspective. www.mathpuzzle.com /groups.html   (581 words)

 [No title]   (Site not responding. Last check: 2007-10-20) Abstract: All finite groups are made up of simple ones, much like the way that all integers are made up of primes. The classification of all finite simple groups, which started with Galois (who introduced this notion, as well as the notion of groups in general), was completed during the second half of the 20th century; this is possibly the most difficult theorem in mathematics. for Lie groups, algebraic groups, or groups of finite Morley rank), seems to me to be one of the main challenges of mathematics for the 21st century. www.math.technion.ac.il /~techm/20040513171520040513fin   (268 words)

 Buecher Kaufempfehlung: The Classification of the Finite Simple Groups: 6 (Mathematical Surveys and Monographs) von ...   (Site not responding. Last check: 2007-10-20) This volume contains the proofs of Theorems C2 and C3, which constitute the classification of finite simple groups G of special odd type. The special odd condition is introduced as representing the measure of smallness for simple groups which are not of even type. The classification of finite simple groups is a landmark result of modern mathematics. www.buchtipp.schnellsuchmaschine.de /0821827774-The_Classification_of_the_Finite_Simple_Groups_6_Mathematical_Surveys_and_Monographs_von_Daniel_Gorenstein_bei_Amer_Mathematical_Society.html   (283 words)

 Steiner system - Wikipedia, the free encyclopedia A finite projective plane of order q can be considered as a Steiner S(2, q+1, q This is connected with many of the sporadic simple groups and with the exceptional 24-dimensional lattice known as the Leech lattice. They form a group under the XOR operation. www.wikipedia.org /wiki/Steiner_system   (416 words)

 Author : works by Daniel Gorenstein The theorem has widespread applications in many branches of applications, as questions about finite groups can often be reduced to questions aboutfinite simple groups, which by the classification can be reduced toan enumeration of cases. Because of the extreme length of the proof of the classification of finitesimple groups, there has been a lot of work (led by Daniel Gorenstein)in trying to find a simpler proof. One reason that some mathematicians believe that a simpler proof ispossible is that the result to be proved is known, which was not the case for the earlier proof. www.bookauthorsearch.com /180295_daniel-gorenstein_0306413051classificationoffinitesimplegroupsgroupsofnoncharacteristic2typebookreport.html   (683 words)

 Probabilistic Generation of Finite Simple Groups - Guralnick, Kantor (ResearchIndex)   (Site not responding. Last check: 2007-10-20) Abstract: For each finite simple group G there is a conjugacy class CG such that each nontrivial element of G generates G together with any of more than 1/10 of the members of CG. Introduction For any finite group G, let PC(G) denote the following probability: PC(G) = max 16=s2G min 16=g2G Prfs 0 2 s G ; hg; s 0 i = Gg: Thus, for at least one conjugacy class CG = s G, PC(G) is a lower bound for the... 102 Atlas of finite groups (context) - Conway, Curtis et al. citeseer.ist.psu.edu /141143.html   (897 words)

 Classification of Simple K*-Groups of Finite (ResearchIndex)   (Site not responding. Last check: 2007-10-20) 5 On groups of finite Morley rank with weakly embedded subgrou.. 3 An identification theorem for groups of finite Morley rank a.. 1 An analog of a theorem of Stellmacher for groups of finite M.. citeseer.ist.psu.edu /614026.html   (429 words)

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