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Thin Finite Group
In the mathematical classification of finite simple groups, a thin group is a finite group such that for every odd prime number ''p'', the Sylow ''p''-subgroups of the 2-local subgroups are cyclic. Informally, these are the groups that resemble rank 1 groups of Lie type over a finite field of characteristic 2. defined thin groups and classified those of characteristic 2 type in which all 2-local subgroups are solvable. The thin simple groups were classified by . The list of finite simple thin groups consists of: *The projective special linear groups PSL2(''q'') and PSL3(''p'') for ''p'' = 1 + 2''a''3''b'' and PSL3(4) *The projective special unitary groups PSU3(''p'') for ''p'' =−1 + 2''a''3''b'' and ''b'' = 0 or 1 and PSU3(2''n'') *The Suzuki groups Sz(2''n'') *The Tits group 2''F''4(2)' *The Steinberg group 3''D''4(2) *The Mathieu group ''M''11 *The Janko group J1 See also *Quasithin group References * * *{{Citation , l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classification Of Finite Simple Groups
In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating groups, alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic groups, sporadic (the Tits group is sometimes regarded as a sporadic group because it is not strictly a group of Lie type, in which case there would be 27 sporadic groups). The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suzuki Groups
In the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(22''n''+1), 2''B''2(22''n''+1), Suz(22''n''+1), or ''G''(22''n''+1), form an infinite family of groups of Lie type found by , that are simple for ''n'' ≥ 1. These simple groups are the only finite non-abelian ones with orders not divisible by 3. Constructions Suzuki originally constructed the Suzuki groups as subgroups of SL4(F22''n''+1) generated by certain explicit matrices. Ree Ree observed that the Suzuki groups were the fixed points of exceptional automorphisms of some symplectic groups of dimension 4, and used this to construct two further families of simple groups, called the Ree groups. In the lowest case the symplectic group ''B''2(2)≈''S''6; its exceptional automorphism fixes the subgroup Sz(2) or 2''B''2(2), of order 20. gave a detailed exposition of Ree's observation. Tits constructed the Suzuki groups as the symmetries of a certain ovoid in 3-dimensional projective sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *'' Memoirs of the American Mathematical Society'' *'' Notices of the American Mathematical Society'' *'' Proceedings of the Ame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. The classification of quasithin groups is a crucial part of the classification of finite simple groups. 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 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 pr ... [...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 (group theory), order : 175,560 = 233571119 : ≈ 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 Arnott Wilson, 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 pariah group, pariahs. Properties The smallest faithful complex representation of ''J1'' has dimension 56. ''J1'' can be characterized abstractly as the unique simple group with abelian Sylow theorems, 2-Sylow subgroups and with an Involution (mathematics), involution whose centralizer is isomorphic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 objects. They are the first sporadic groups to be discovered. Sometimes the notation ''M''8, ''M''9, ''M''10, ''M''20, and ''M''21 is used for related groups (which act on sets of 8, 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 the projective special linear group PSL(3,4). History introduced the group ''M''12 as part of an investigation of multiply transitive permutat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinberg Group (Lie Theory)
In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase ''group of Lie type'' does not have a widely accepted precise definition, but the important collection of finite simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups. The name "groups of Lie type" is due to the close relationship with the (infinite) Lie groups, since a compact Lie group may be viewed as the rational points of a reductive linear algebraic group over the field of real numbers. and are standard references for groups of Lie type. Classical groups An initial approach to this question was the definition and detailed study of the so-called ''classical groups'' over finite and other fields by . These groups were studied by L. E. Dickson ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Group
In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order : 17,971,200 = 211 · 33 · 52 · 13. This is the only simple group that is a derivative of a group of Lie type that is not a group of Lie type in any series from exceptional isomorphisms. It is sometimes considered a 27th sporadic group. History and properties The Ree groups 2''F''4(22''n''+1) were constructed by , who showed that they are simple if ''n'' ≥ 1. The first member 2''F''4(2) of this series is not simple. It was studied by who showed that it is almost simple, its derived subgroup 2''F''4(2)′ of index 2 being a new simple group, now called the Tits group. The group 2''F''4(2) is a group of Lie type and has a BN pair, but the Tits group itself does not have a BN pair. The Tits group is member of the infinite family 2''F''4(22''n''+1)′ of commutator groups of the Ree groups, and thus by def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Group
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Finite Group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod n, integers mod p when p is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set that is a fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Of Lie Type
In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a Reductive group, reductive linear algebraic group with values in a finite field. The phrase ''group of Lie type'' does not have a widely accepted precise definition, but the important collection of finite Simple group, simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups. The name "groups of Lie type" is due to the close relationship with the (infinite) Lie groups, since a compact Lie group may be viewed as the rational points of a reductive linear algebraic group over the field of real numbers. and are standard references for groups of Lie type. Classical groups An initial approach to this question was the definition and detailed study of the so-called ''classical groups'' over finite and other field (mathematics), fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
