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Pariah Group
In group theory, the term pariah was introduced by Robert Griess in to refer to the six sporadic simple groups which are not subquotients of the monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 246320597611213317192329314147 .... The twenty groups which are subquotients, including the monster group itself, he dubbed the happy family. For example, the orders of ''J''4 and the Lyons Group ''Ly'' are divisible by 37. Since 37 does not divide the order of the monster, these cannot be subquotients of it; thus ''J''4 and ''Ly'' are pariahs. Three other sporadic groups were also shown to be pariahs by Griess in 1982, and the Janko Group J1 was shown to be the final pariah by Robert A. Wilson in 1986. The complete list is shown below. References * * Robert A. Wilson (1986)''Is J1 a subgroup of the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Griess
Robert Louis Griess, Jr. (born 1945, Savannah, Georgia) is a mathematician working on finite simple groups and vertex algebras. He is currently the John Griggs Thompson Distinguished University Professor of mathematics at University of Michigan. Education Griess developed a keen interest in mathematics prior to entering undergraduate studies at the University of Chicago in the fall of 1963. There, he eventually earned a Ph.D. in 1971 under the supervision of John Griggs Thompson after defending a dissertation on the Schur multipliers of the then-known finite simple groups. Research Griess' early work focused on group extensions, cohomology and Schur multipliers, as well as on the classification of finite simple groups and their properties. He was awarded a Guggenheim Fellowship in 1981. In 1982, he published the first construction of the monster group using the Griess algebra, for which he was awarded the 2010 AMS Leroy P. Steele Prize for Seminal Contribution to Research. He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sporadic Simple Groups
In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group, in which case there would be 27 sporadic groups. The monster group is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it. Names Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subquotient
In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with a different meaning in category theory. In the literature about sporadic groups wordings like «H is involved in G» can be found with the apparent meaning of «H is a subquotient of G». A quotient of a subrepresentation of a representation (of, say, a group) might be called a subquotient representation; e.g., Harish-Chandra's subquotient theorem. p. 310 Examples Of the 26 sporadic groups, the 20 subquotients of the monster group are referred to as the "Happy Family", whereas the remaining 6 as "pariah groups". Order relation The relation ''subquotient of'' is an order relation. Proof of transitivity for groups Let H'/H'' be subquotient of H, furthermore H := G'/G'' be subquotient of G and \varphi \colon G' \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monster Group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 2463205976112133171923293141475971 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 ≈ 8. The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as subquotients. Robert Griess, who proved the existence of the monster in 1982, has called those 20 groups the ''happy family'', and the remaining six exceptions ''pariahs''. It is difficult to give a good constructive definition of the monster because of its complexity. Martin Gardner wrote a popular account of the monster group in his June 1980 Mathematical Games column in ''Scientific ... [...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 A5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Arnott Wilson
Robert Arnott Wilson (born 1958) is a retired mathematician in London, England, who is best known for his work on classifying the maximal subgroups of finite simple groups and for the work in the Monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 246320597611213317192329314147 .... He is also an accomplished violin, viola and piano player, having played as the principal viola in the Sinfonia of Birmingham. Due to a damaged finger, he now principally plays the kora. Books * *''An Atlas of Brauer Characters'' (London Mathematical Society Monographs) by Christopher Jansen, Klaus Lux, Richard Parker, Robert Wilson. Oxford University Press, USA (October 1, 1995) * as editor * Selected articles * * with Peter B. Kleidman: * with R. A. Parker: * with M. D. E. Conder and A. J. Woldar: * * * * * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyons Group
In the area of modern algebra known as group theory, the Lyons group ''Ly'' or Lyons-Sims group ''LyS'' is a sporadic simple group of order : 283756711313767 : = 51765179004000000 : ≈ 5. History ''Ly'' is one of the 26 sporadic groups and was discovered by Richard Lyons and Charles Sims in 1972-73. Lyons characterized 51765179004000000 as the unique possible order of any finite simple group where the centralizer of some involution is isomorphic to the nontrivial central extension of the alternating group A11 of degree 11 by the cyclic group C2. proved the existence of such a group and its uniqueness up to isomorphism with a combination of permutation group theory and machine calculations. When the McLaughlin sporadic group was discovered, it was noticed that a centralizer of one of its involutions was the perfect double cover of the alternating group ''A''8. This suggested considering the double covers of the other alternating groups ''A''''n'' as possi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
O'Nan Group
In the area of abstract algebra known as group theory, the O'Nan group ''O'N'' or O'Nan–Sims group is a sporadic simple group of order : 2934573111931 : = 460815505920 : ≈ 5. History ''O'Nan'' is one of the 26 sporadic groups and was found by in a study of groups with a Sylow 2-subgroup of " Alperin type", meaning isomorphic to a Sylow 2-Subgroup of a group of type (Z/2''n''Z ×Z/2''n''Z ×Z/2''n''Z).PSL3(F2). For the O'Nan group ''n'' = 2 and the extension does not split. The only other simple group with a Sylow 2-subgroup of Alperin type with ''n'' ≥ 2 is the Higman–Sims group again with ''n'' = 2, but the extension splits. The Schur multiplier has order 3, and its outer automorphism group has order 2. showed that O'Nan cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs. Representations showed that its triple cover has two 45-dimensional representations over the field with 7 elements, excha ... [...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 the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Janko Group J4
In the area of modern algebra known as group theory, the Janko group ''J4'' is a sporadic simple group of order : 22133571132329313743 : = 86775571046077562880 : ≈ 9. History ''J4'' is one of the 26 Sporadic groups. Zvonimir Janko found J4 in 1975 by studying groups with an involution centralizer of the form 21 + 12.3.(M22:2). Its existence and uniqueness was shown using computer calculations by Simon P. Norton and others in 1980. It has a modular representation of dimension 112 over the finite field with 2 elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. and gave computer-free proofs of uniqueness. and gave a computer-free proof of existence by constructing it as an amalgams of groups 210:SL5(2) and (210:24:A8):2 over a group 210:24:A8. The Schur multiplier and the outer automorphism group are both tri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
