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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] |
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] |
Trivial Group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: 0, 1, or e depending on the context. If the group operation is denoted \, \cdot \, then it is defined by e \cdot e = e. The similarly defined is also a group since its only element is its own inverse, and is hence the same as the trivial group. The trivial group is distinct from the empty set, which has no elements, hence lacks an identity element, and so cannot be a group. Definitions Given any group G, the group consisting of only the identity element is a subgroup of G, and, being the trivial group, is called the of G. The term, when referred to "G has no nontrivial proper subgroups" refers to the only subgroups of G being the trivial group \ and the group G itself. Properties The trivial group is cyclic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg Group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ''a, b'' and ''c'' can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group"). The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to ''n''-dimensional systems, and most generally, to any symplectic vector space. The three-dimensional case In the three-dimensional case, the product of two Heisenberg matrices is given by: :\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end \begin 1 & a' & c'\\ 0 & 1 & b'\\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
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] |
Supersingular Prime (moonshine Theory)
In the mathematical branch of moonshine theory, a supersingular prime is a prime number that divides the order of the Monster group ''M'', which is the largest sporadic simple group. There are precisely fifteen supersingular prime numbers: the first eleven primes ( 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31), as well as 41, 47, 59, and 71. The non-supersingular primes are 37, 43, 53, 61, 67, and any prime number greater than or equal to 73. Supersingular primes are related to the notion of supersingular elliptic curves as follows. For a prime number ''p'', the following are equivalent: # The modular curve ''X''0+(''p'') = ''X''0(''p'') / ''w''p, where ''w''p is the Fricke involution of ''X''0(''p''), has genus zero. # Every supersingular elliptic curve in characteristic ''p'' can be defined over the prime subfield F''p''. # The order of the Monster group is divisible by ''p''. The equivalence is due to Andrew Ogg. More precisely, in 1975 Ogg showed that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 trivial center, then is said to be complete. An automorphism of a group which is not inner is called an outer automorphism. The cosets of with respect to outer automorphisms are then the elements of ; this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the automorphism group and outer automorphism group are naturally identified; that is, the outer automorphism group does act on the group. For example, for the alternating group, , the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering as a subgroup of the symmetric group, , conjugation by any odd permutation is an oute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sporadic Simple Group
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 exis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \operatorname(G) of a finite group ''G'' is a finite abelian group whose exponent divides the order of ''G''. If a Sylow ''p''-subgroup of ''G'' is cyclic for some ''p'', then the order of \operatorname(G) is not divisible by ''p''. In particular, if all Sylow ''p''-subgroups of ''G'' are cyclic, then \operatorname(G) is trivial. For instance, the Schur multiplier of the nonabelian group of order 6 is the trivial group since every Sylow subgroup is cyclic. The Schur multiplier of the elementary abelian group of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. The Schur multiplier of the quaternion group is trivial, but the Schur multiplier of dihedral 2-groups ... [...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 that contains a finite number of elements. As with any field, a finite field is a 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 given by the integers mod when 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 and every positive integer there are fields of order p^k, all of which are 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 which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
