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Sporadic 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 exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Group M23
In the area of modern algebra known as group theory, the Mathieu group ''M''23 is a sporadic simple group of order : 2732571123 = 10200960 : ≈ 1 × 107. History and properties ''M''23 is one of the 26 sporadic groups and was introduced by . It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial. calculated the integral cohomology, and showed in particular that M23 has the unusual property that the first 4 integral homology groups all vanish. The inverse Galois problem seems to be unsolved for M23. In other words, no polynomial in Z 'x''seems to be known to have M23 as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups. Construction using finite fields Let be the finite field with 211 elements. Its group of units has order − 1 = 2047 = 23 · 89, so it has a cyclic subgroup of order 23. The Mathieu group M23 can be identified with ... [...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 : 21831352711131723 : = 4089470473293004800 : ≈ 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 The smallest faithful complex representation has dimension 782. The group has an irreducible representation of dimension 253 over the field with 3 elements. Generalized Monstrous Moonshine Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fischer Group Fi22
In the area of modern algebra known as group theory, the Fischer group ''Fi22'' is a sporadic simple group of order : 217395271113 : = 64561751654400 : ≈ 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 real representation of dimension 78. Reducing an integral form of this mod 3 gives a representation of Fi22 over the field with 3 elements, whose quotient by the 1-dimensional space of fixed vectors is a 77-dimensional irreducible representation. The perfect tr ... [...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 : 210375371123 : = 495766656000 : ≈ 5. History and properties ''\mathrm_3'' is one of the 26 sporadic groups and was discovered by as the group of automorphisms of the Leech lattice \Lambda fixing a lattice vector of type 3, thus length . It is thus a subgroup of \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. Representations Co3 acts on the unique 23-dimensional even lattice of determinant 4 with no roots, given by the orthogonal complement of a norm 4 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 doubly transitive permutation representation ... [...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 : 218365371123 : = 42305421312000 : ≈ 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 In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ... are both Trivial group, 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 act ... [...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 : 221395472111323 : = 4157776806543360000 : ≈ 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 4-eleme ... [...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 : 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 bot ... [...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 : 273551719 = 50232960. 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 eighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Janko Group J2
In the area of modern algebra known as group theory, the Janko group ''J2'' or the Hall-Janko group ''HJ'' is a sporadic simple group of order : 2733527 = 604800 : ≈ 6. History and properties ''J2'' is one of the 26 Sporadic groups and is also called Hall–Janko–Wales group. In 1969 Zvonimir Janko predicted J2 as one of two new simple groups having 21+4:A5 as a centralizer of an involution (the other is the Janko group J3). It was constructed by as a rank 3 permutation group on 100 points. Both the Schur multiplier and the outer automorphism group have order 2. As a permutation group on 100 points J2 has involutions moving all 100 points and involutions moving just 80 points. The former involutions are products of 25 double transportions, an odd number, and hence lift to 4-elements in the double cover 2.A100. The double cover 2.J2 occurs as a subgroup of the Conway group Co0. J2 is the only one of the 4 Janko groups that is a subquotient of the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
