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Sporadic Groups
In the mathematical classification of finite simple groups, there are a number of Group (mathematics), groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The mentioned classification theorem states that the list of finite simple groups consists of 18 countably infinite families plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. 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 monster group, or ''friendly giant'', 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 Émile Léonard Math ... [...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] |
Mathieu Group M24
In the area of modern algebra known as group theory, the Mathieu group ''M24'' is a sporadic simple group of order : 244,823,040 = 21033571123 : ≈ 2. History and properties ''M24'' is one of the 26 sporadic groups and was introduced by . It is a 5-transitive permutation group on 24 objects. The Schur multiplier and the outer automorphism group are both trivial. The Mathieu groups can be constructed in various ways. Initially, Mathieu and others constructed them as permutation groups. It was difficult to see that M24 actually existed, that its generators did not just generate the alternating group A24. The matter was clarified when Ernst Witt constructed M24 as the automorphism (symmetry) group of an S(5,8,24) Steiner system W24 (the Witt design). M24 is the group of permutations that map every block in this design to some other block. The subgroups M23 and M22 then are easily defined to be the stabilizers of a single point and a pair of points res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fischer Group Fi24
In the area of modern algebra known as group theory, the Fischer group ''Fi24'' or ''F24'' or ''F3+'' is a sporadic simple group of order : 1,255,205,709,190,661,721,292,800 : = 22131652731113172329 : ≈ 1. History and properties ''Fi24'' is one of the 26 sporadic groups and is the largest of the three Fischer groups introduced by while investigating 3-transposition groups. It is the 3rd largest of the sporadic groups (after the Monster group and Baby Monster group). The outer automorphism group has order 2, and the Schur multiplier has order 3. The automorphism group is a 3-transposition group Fi24, containing the simple group with index 2. The centralizer of an element of order 3 in the monster group is a triple cover of the sporadic simple group ''Fi24'', as a result of which the prime 3 plays a special role in its theory. Representations The centralizer of an element of order 3 in the monster group In the area of abstract algebra known as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fischer Group Fi23
In the area of modern algebra known as group theory, the Fischer group ''Fi23'' is a sporadic simple group of order : 4,089,470,473,293,004,800 : = 21831352711131723 : ≈ 4. History ''Fi23'' is one of the 26 sporadic groups and is one of the three Fischer groups introduced by while investigating 3-transposition groups. The Schur multiplier and the outer automorphism group are both trivial. Representations The Fischer group Fi23 has a rank 3 action on a graph of 31671 vertices corresponding to 3-transpositions, with point stabilizer the double cover of the Fischer group Fi22. It has a second rank-3 action on 137632 points Fi23 is the centralizer of a transposition in the Fischer group Fi24. When realizing Fi24 as a subgroup 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; it has order :&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fischer Group Fi22
In the area of modern algebra known as 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 ( ..., the Fischer group ''Fi22'' is a sporadic simple group of order : 64,561,751,654,400 : = 217395271113 : ≈ 6. History ''Fi22'' is one of the 26 sporadic groups and is the smallest of the three Fischer groups. It was introduced by while investigating 3-transposition groups. The outer automorphism group has order 2, and the Schur multiplier has order 6. Representations The Fischer group Fi22 has a rank 3 action on a graph of 3510 vertices corresponding to its 3-transpositions, with point stabilizer the double cover of the group PSU6(2). It also has two rank 3 actions on 14080 points, exchanged by an outer automorphism. Fi22 has an irreducible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Conway Group Co3
In the area of modern algebra known as group theory, the Conway group ''\mathrm_3'' is a sporadic simple group of Order (group theory), order : 495,766,656,000 : = 210375371123 : ≈ 5. History and properties ''\mathrm_3'' is one of the 26 sporadic groups and was discovered by as the Automorphism group, group of automorphisms of the Leech lattice \Lambda fixing a lattice vector of type 3, thus length . It is thus a subgroup of Conway group, \mathrm_0. It is isomorphic to a subgroup of \mathrm_1. The direct product 2\times \mathrm_3 is maximal in \mathrm_0. The Schur multiplier and the outer automorphism group are both Trivial group, trivial. Representations Co3 acts on a 23-dimensional even lattice with no roots, given by the orthogonal complement of a norm 6 vector of the Leech lattice. This gives 23-dimensional representations over any field; over fields of characteristic 2 or 3 this can be reduced to a 22-dimensional faithful representation. Co3 has a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conway Group Co2
In the area of modern algebra known as group theory, the Conway group ''Co2'' is a sporadic simple group of order : 42,305,421,312,000 : = 218365371123 : ≈ 4. History and properties ''Co2'' is one of the 26 sporadic groups and was discovered by as the group of automorphisms of the Leech lattice Λ fixing a lattice vector of type 2. It is thus a subgroup of Co0. It is isomorphic to a subgroup of Co1. The direct product 2×Co2 is maximal in Co0. The Schur multiplier and the outer automorphism group are both trivial. Representations Co2 acts as a rank 3 permutation group on 2300 points. These points can be identified with planar hexagons in the Leech lattice having 6 type 2 vertices. Co2 acts on the 23-dimensional even integral lattice with no roots of determinant 4, given as a sublattice of the Leech lattice orthogonal to a norm 4 vector. Over the field with 2 elements it has a 22-dimensional faithful representation; this is the smallest faithful r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conway Group Co1
In the area of modern algebra known as group theory, the Conway group ''Co1'' is a sporadic simple group of order : 4,157,776,806,543,360,000 : = 221395472111323 : ≈ 4. History and properties ''Co1'' is one of the 26 sporadic groups and was discovered by John Horton Conway in 1968. It is the largest of the three sporadic Conway groups and can be obtained as the quotient of ''Co0'' ( group of automorphisms of the Leech lattice Λ that fix the origin) by its center, which consists of the scalar matrices ±1. It also appears at the top of the automorphism group of the even 26-dimensional unimodular lattice II25,1. Some rather cryptic comments in Witt's collected works suggest that he found the Leech lattice and possibly the order of its automorphism group in unpublished work in 1940. The outer automorphism group is trivial and the Schur multiplier has order 2. Involutions Co0 has 4 conjugacy classes of involutions; these collapse to 2 in Co1, but there are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Janko Group J4
In the area of modern algebra known as group theory, the Janko group ''J4'' is a sporadic simple group of order : 86,775,571,046,077,562,880 : = 22133571132329313743 : ≈ 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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Janko Group J3
In the area of modern algebra known as group theory, the Janko group ''J3'' or the Higman-Janko-McKay group ''HJM'' is a sporadic simple group of order : 50,232,960 = 273551719. History and properties ''J3'' is one of the 26 Sporadic groups and was predicted by Zvonimir Janko in 1969 as one of two new simple groups having 21+4:A5 as a centralizer of an involution (the other is the Janko group ''J2''). ''J3'' was shown to exist by . In 1982 R. L. Griess showed that ''J3'' cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs. J3 has an outer automorphism group of order 2 and a Schur multiplier of order 3, and its triple cover has a unitary 9-dimensional representation over the finite field with 4 elements. constructed it via an underlying geometry. It has a modular representation of dimension eighteen over the finite field with 9 elements. It has a complex projective representation of dimension eigh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
