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Finite Simple Group
In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates. Summary The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families. (In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A8 = ''A''3(2) and ''A''2(4) both have order 20160, and that the group ''Bn''(''q'') has the same order as ''Cn''(''q'') for ''q'' odd, ''n'' > 2. The smalle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 : 51,765,179,004,000,000 : = 283756711313767 : ≈ 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Harada–Norton Group
In the area of modern algebra known as group theory, the Harada–Norton group ''HN'' is a sporadic simple group of order : 273,030,912,000,000 : = 214365671119 : ≈ 3. History and properties ''HN'' is one of the 26 sporadic groups and was found by and ). Its Schur multiplier is trivial and its outer automorphism group has order 2. ''HN'' has an involution whose centralizer is of the form 2.HS.2, where HS is the Higman-Sims group (which is how Harada found it). The prime 5 plays a special role in the group. For example, it centralizes an element of order 5 in the Monster group (which is how Norton found it), and as a result acts naturally on a vertex operator algebra over the field with 5 elements . This implies that it acts on a 133 dimensional algebra over F5 with a commutative but nonassociative product, analogous to the Griess algebra . The full normalizer of a 5A element in the Monster group is (D10 × HN).2, so HN centralizes 5 involutions alongside ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 : 460,815,505,920 = 2934573111931 ≈ 5. History ''O'N'' 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). The following simple groups have Sylow 2-subgroups of Alperin type: * For the Chevalley group ''G''2(q), if q is congruent to 3 or 5 mod 8, ''n = 1'' and the extension does not split. * For the Steinberg group 3''D''4(q), if q is congruent to 3 or 5 mod 8, ''n = 1'' and the extension does not split. * For the alternating group A8, ''n = 1'' and the extension splits. * For the O'Nan group, ''n'' = 2 and the extension does not split. * For the Higman-Sims group, ''n'' = 2 and the extension splits. The Schur multiplier has order 3, and its outer a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Suzuki Sporadic Group
In the area of modern algebra known as group theory, the Suzuki group ''Suz'' or ''Sz'' is a sporadic simple group of order : 448,345,497,600 = 213 · 37 · 52 · 7 · 11 · 13 ≈ 4. History ''Suz'' is one of the 26 Sporadic groups and was discovered by as a rank 3 permutation group on 1782 points with point stabilizer G2(4). It is not related to the Suzuki groups of Lie type. The Schur multiplier has order 6 and the outer automorphism group has order 2. Complex Leech lattice The 24-dimensional Leech lattice has a fixed-point-free automorphism of order 3. Identifying this with a complex cube root of 1 makes the Leech lattice into a 12 dimensional lattice over the Eisenstein integers In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form : z = a + b\omega , where and are integers and : \omega = \frac ..., called the complex Leech ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rudvalis Group
In the area of modern algebra known as group theory, the Rudvalis group ''Ru'' is a sporadic simple group of order : 145,926,144,000 = 214335371329 : ≈ 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, 1280). 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 1280 . Its double cover acts on a 28-dimensional lattice over t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Held Group
In the area of modern algebra known as group theory, the Held group ''He'' is a sporadic simple group of order : 4,030,387,200 = 21033527317 : ≈ 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 an extension of the extra special group 21+6 by the linear group L3(2), which is the same involution centralizer as 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. In all of these groups, the extension splits. 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 spec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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McLaughlin Group (mathematics)
In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order : 898,128,000 = 27 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 : ≈ 9. History and properties McL is one of the 26 sporadic groups and was discovered by as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with vertices. It fixes a 2-2-3 triangle in the Leech lattice and thus is a subgroup of the Conway groups \mathrm_0, \mathrm_2, and \mathrm_3. Its Schur multiplier has order 3, and its outer automorphism group has order 2. The group 3.McL:2 is a maximal subgroup of the Lyons group. McL has one conjugacy class of involution (element of order 2), whose centralizer is a maximal subgroup of type 2.A8. This has a center of order 2; the quotient modulo the center is isomorphic to the alternating group A8. Representations In the Conway group Co3, McL has the normalizer McL:2, which is maximal in Co3. McL has 2 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Higman–Sims Group
In the area of modern algebra known as group theory, the Higman–Sims group HS is a sporadic simple group of order : 44,352,000 = 293253711 : ≈ 4. The Schur multiplier has order 2, the outer automorphism group has order 2, and the group 2.HS.2 appears as an involution centralizer in the Harada–Norton group. History HS is one of the 26 sporadic groups and was found by . They were attending a presentation by Marshall Hall on the Hall–Janko group J2. It happens that J2 acts as a permutation group on the Hall–Janko graph of 100 points, the stabilizer of one point being a subgroup with two other orbits of lengths 36 and 63. Inspired by this they decided to check for other rank 3 permutation groups on 100 points. They soon focused on a possible one containing the Mathieu group M22, which has permutation representations on 22 and 77 points. (The latter representation arises because the M22 Steiner system has 77 blocks.) By putting together these two represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fischer Group
In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by . 3-transposition groups The Fischer groups are named after Bernd Fischer who discovered them while investigating 3-transposition groups. These are groups ''G'' with the following properties: * ''G'' is generated by a conjugacy class of elements of order 2, called 'Fischer transpositions' or 3-transpositions. * The product of any two distinct transpositions has order 2 or 3. The typical example of a 3-transposition group is a symmetric group, where the Fischer transpositions are genuinely transpositions. The symmetric group Sn can be generated by transpositions: (12), (23), ..., . Fischer was able to classify 3-transposition groups that satisfy certain extra technical conditions. The groups he found fell mostly into several infinite classes (besides symmetric groups: certain classes of symplectic, unitary, and orthogonal groups) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Conway Group
In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by . The largest of the Conway groups, Co0, is the group of automorphisms of the Leech lattice Λ with respect to addition and inner product. It has order : but it is not a simple group. The simple group Co1 of order : = 221395472111323 is defined as the quotient of Co0 by its center, which consists of the scalar matrices ±1. The groups Co2 of order : = 218365371123 and Co3 of order : = 210375371123 consist of the automorphisms of Λ fixing a lattice vector of type 2 and type 3, respectively. As the scalar −1 fixes no non-zero vector, these two groups are isomorphic to subgroups of Co1. The inner product on the Leech lattice is defined as 1/8 the sum of the products of respective co-ordinates of the two multiplicand vectors; it is an integer. The square norm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |