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McLaughlin Sporadic Group
In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order : 27 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 = 898,128,000 : ≈ 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 cl ... [...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] |
Point Stabilizer
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Algebra
''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1984. From 1985 until 2000, Walter Feit served as its editor-in-chief. In 2004, ''Journal of Algebra'' announced (vol. 276, no. 1 and 2) the creation of a new section on computational algebra, with a separate editorial board. The first issue completely devoted to computational algebra was vol. 292, no. 1 (October 2005). The Editor-in-Chief of the ''Journal of Algebra'' is Michel Broué, Université Paris Diderot, and Gerhard Hiß, Rheinisch-Westfälische Technische Hochschule Aachen ( RWTH) is Editor of the computational algebra section. See also *Susan Montgomery M. Susan Montgomery (born 2 April 1943 in Lansing, MI) is a distinguished American mathematician whose current research interests concern noncommutative algebras: in parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Arnott Wilson
Robert Arnott Wilson (born 1958) is a retired mathematician in London, England, who is best known for his work on classifying the maximal subgroups of finite simple groups and for the work in 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 .... He is also an accomplished violin, viola and piano player, having played as the principal viola in the Sinfonia of Birmingham. Due to a damaged finger, he now principally plays the kora. Books * *''An Atlas of Brauer Characters'' (London Mathematical Society Monographs) by Christopher Jansen, Klaus Lux, Richard Parker, Robert Wilson. Oxford University Press, USA (October 1, 1995) * as editor * Selected articles * * with Peter B. Kleidman: * with R. A. Parker: * with M. D. E. Conder and A. J. Woldar: * * * * * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simon P
Simon may refer to: People * Simon (given name), including a list of people and fictional characters with the given name Simon * Simon (surname), including a list of people with the surname Simon * Eugène Simon, French naturalist and the genus authority ''Simon'' * Tribe of Simeon, one of the twelve tribes of Israel Places * Şimon ( hu, links=no, Simon), a village in Bran Commune, Braşov County, Romania * Șimon, a right tributary of the river Turcu in Romania Arts, entertainment, and media Films * ''Simon'' (1980 film), starring Alan Arkin * ''Simon'' (2004 film), Dutch drama directed by Eddy Terstall Games * ''Simon'' (game), a popular computer game * Simon Says, children's game Literature * ''Simon'' (Sutcliff novel), a children's historical novel written by Rosemary Sutcliff * Simon (Sand novel), an 1835 novel by George Sand * ''Simon Necronomicon'' (1977), a purported grimoire written by an unknown author, with an introduction by a man identified only as "Simon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius College, Camb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Group M11
In the area of modern algebra known as group theory, the Mathieu group ''M11'' is a sporadic simple group of order : 2432511 = 111098 = 7920. History and properties ''M11'' is one of the 26 sporadic groups and was introduced by . It is the smallest sporadic group and, along with the other four Mathieu groups, the first to be discovered. The Schur multiplier and the outer automorphism group are both trivial. ''M11'' is a sharply 4-transitive permutation group on 11 objects. It admits many generating sets of permutations, such as the pair (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) of permutations used by the GAP computer algebra system. Representations M11 has a sharply 4-transitive permutation representation on 11 points. The point stabilizer is sometimes denoted by M10, and is a non-split extension of the form A6.2 (an extension of the group of order 2 by the alternating group A6). This action is the automorphism group of a Steiner system S(4,5,11) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Group M22
In the area of modern algebra known as group theory, the Mathieu group ''M22'' is a sporadic simple group of Order (group theory), order : 27325711 = 443520 : ≈ 4. History and properties ''M22'' is one of the 26 sporadic groups and was introduced by . It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M22 is cyclic of order 12, and the outer automorphism group has order 2. There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature. incorrectly claimed that the Schur multiplier of M22 has order 3, and in a correction incorrectly claimed that it has order 6. This caused an error in the title of the paper announcing the discovery of the Janko group J4. showed that the Schur multiplier is in fact cyclic of order 12. calculated the 2-part of all the cohomology of M22. Representations M22 has a 3-transitive permutation representation on 22 points, with point stabilizer th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternionic Representation
In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space ''V'' with an invariant quaternionic structure, i.e., an antilinear equivariant map :j\colon V\to V which satisfies :j^2=-1. Together with the imaginary unit ''i'' and the antilinear map ''k'' := ''ij'', ''j'' equips ''V'' with the structure of a quaternionic vector space (i.e., ''V'' becomes a module over the division algebra of quaternions). From this point of view, quaternionic representation of a group ''G'' is a group homomorphism ''φ'': ''G'' → GL(''V'', H), the group of invertible quaternion-linear transformations of ''V''. In particular, a quaternionic matrix representation of ''g'' assigns a square matrix of quaternions ''ρ''(g) to each element ''g'' of ''G'' such that ''ρ''(e) is the identity matrix and :\rho(gh)=\rho(g)\rho(h)\textg, h \in G. Quaternionic representations of associative and Lie algebras can ... [...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] |
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 were the first sporadic groups to be discovered. Sometimes the notation ''M''9, ''M''10, ''M''20 and ''M''21 is used for related groups (which act on sets of 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 PSL(3,4). History introduced the group ''M''12 as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
