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Quasithin Group
In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an abelian group of odd order normalizing a non-trivial 2-subgroup of ''G''. When ''G'' is a group of Lie type of characteristic 2 type, the width is usually the rank (the dimension of a maximal torus of the algebraic group). Classification The classification of quasithin groups is a crucial part of the classification of finite simple groups. The quasithin groups were classified in a 1221-page paper by . An earlier announcement by of the classification, on the basis of which the classification of finite simple groups was announced as finished in 1983, was premature as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic properties For , the group A''n'' is the commutator subgroup of the symmetric group S''n'' with index 2 and has therefore ''n''!/2 elements. It is the kernel of the signature group homomorphism explained under symmetric group. The group A''n'' is abelian if and only if and simple if and only if or . A5 is the smallest non-abelian simple group, having order 60, and the smallest non- solvable group. The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto . We have the exact sequence . In Galois theory, this map, or rather the corresponding map , corresponds to associating the Lagrange resolvent cubic to a quartic, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Janko Group J1
In the area of modern algebra known as group theory, the Janko group ''J1'' is a sporadic simple group of order : 233571119 = 175560 : ≈ 2. History ''J1'' is one of the 26 sporadic groups and was originally described by Zvonimir Janko in 1965. It is the only Janko group whose existence was proved by Janko himself and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century. Its discovery launched the modern theory of sporadic groups. In 1986 Robert A. Wilson showed that ''J1'' cannot be a subgroup of the monster group. Thus it is one of the 6 sporadic groups called the pariahs. Properties The smallest faithful complex representation of ''J1'' has dimension 56.Jansen (2005), p.123 ''J1'' can be characterized abstractly as the unique simple group with abelian 2-Sylow subgroups and with an involution whose centralizer is isomorphic to the direct product of the group of order two and the alternating group A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald Solomon
Ronald "Ron“ Mark Solomon (b. 15 December 1948 ) is an American mathematician specializing in the theory of . Solomon studied as an undergraduate at Queens College and received a PhD in 1971 at under Walter Feit with a thesis entitled ''Finite Groups with Sylow 2-Subgr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Lyons (mathematician)
Richard Neil Lyons (born January 22, 1945 in New York City, New York) is an American mathematician, specializing in finite group theory. Lyons received his PhD in 1970 at the University of Chicago under John Griggs Thompson with a thesis entitled ''Characterizations of Some Finite Simple Groups with Small 2-Rank''. From 1972 to 2017, he was a professor at Rutgers University. With Daniel Gorenstein and Ronald Solomon he wrote, and is continuing to write, a series on the classification program for finite simple groups, a program in which the three of them were major participants. Nine volumes of this series have been published so far. He discovered a sporadic group which Charles Sims constructed and called the Lyons group ''Ly''. In 2012, he shared the Leroy P. Steele Prize for Mathematical Exposition, awarded by the American Mathematical Society, with Michael Aschbacher, Stephen D. Smith, and Ronald Solomon. In 2013, he became a fellow of the American Mathematical Society ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Gorenstein
Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertation a duality principle for plane curves that motivated Grothendieck's introduction of Gorenstein rings. He was a major influence on the classification of finite simple groups. After teaching mathematics to military personnel at Harvard before earning his doctorate, Gorenstein held posts at Clark University and Northeastern University before he began teaching at Rutgers University in 1969, where he remained for the rest of his life. He was the founding director of DIMACS in 1989, and remained as its director until his death. Charles Weibel. Gorenst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rudvalis Group
In the area of modern algebra known as group theory, the Rudvalis group ''Ru'' is a sporadic simple group of order : 214335371329 : = 145926144000 : ≈ 1. History ''Ru'' is one of the 26 sporadic groups and was found by and constructed by . Its Schur multiplier has order 2, and its outer automorphism group is trivial. In 1982 Robert Griess showed that ''Ru'' cannot be a subquotient of the monster group.Griess (1982) Thus it is one of the 6 sporadic groups called the pariahs. Properties The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer being the Ree group 2''F''4(2), the automorphism group of the Tits group. This representation implies a strongly regular graph srg(4060, 2304, 1328, 1208). That is, each vertex has 2304 neighbors and 1755 non-neighbors, any two adjacent vertices have 1328 common neighbors, while any two non-adjacent ones have 1208 . Its double cover acts on a 28-dimensional lattice over th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Held Group
In the area of modern algebra known as group theory, the Held group ''He'' is a sporadic simple group of order : 21033527317 = 4030387200 : ≈ 4. History ''He'' is one of the 26 sporadic groups and was found by during an investigation of simple groups containing an involution whose centralizer is isomorphic to that of an involution in the Mathieu group M24. A second such group is the linear group L5(2). The Held group is the third possibility, and its construction was completed by John McKay and Graham Higman. The outer automorphism group has order 2 and the Schur multiplier is trivial. Representations The smallest faithful complex representation has dimension 51; there are two such representations that are duals of each other. It centralizes an element of order 7 in the Monster group. As a result the prime 7 plays a special role in the theory of the group; for example, the smallest representation of the Held group over any field is the 50-dimens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Janko Group
In the area of modern algebra known as group theory, the Janko groups are the four sporadic simple groups '' J1'', '' J2'', '' J3'' and '' J4'' introduced by Zvonimir Janko. Unlike the Mathieu groups, Conway groups, or Fischer groups, the Janko groups do not form a series, and the relation among the four groups is mainly historical rather than mathematical. History Janko constructed the first of these groups, ''J''1, in 1965 and predicted the existence of ''J''2 and ''J''3. In 1976, he suggested the existence of ''J''4. Later, ''J''2, ''J''3 and ''J''4 were all shown to exist. ''J''1 was the first sporadic simple group discovered in nearly a century: until then only the Mathieu group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 obje ...s were known, ''M''11 and ''M''12 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Groups
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They were the first sporadic groups to be discovered. Sometimes the notation ''M''9, ''M''10, ''M''20 and ''M''21 is used for related groups (which act on sets of 9, 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones. John Conway has shown that one can also extend this sequence up, obtaining the Mathieu groupoid ''M''13 acting on 13 points. ''M''21 is simple, but is not a sporadic group, being isomorphic to PSL(3,4). History introduced the group ''M''12 as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
