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O'Nan–Scott Theorem
In mathematics, the O'Nan–Scott theorem is one of the most influential theorems of permutation group theory; the classification of finite simple groups is what makes it so useful. Originally the theorem was about maximal subgroups of the symmetric group. It appeared as an appendix to a paper by Leonard Scott written for The Santa Cruz Conference on Finite Groups in 1979, with a footnote that Michael O'Nan had independently proved the same result. Michael Aschbacher and Scott later gave a corrected version of the statement of the theorem. The theorem states that a maximal subgroup of the symmetric group Sym(Ω), where , Ω, = ''n'', is one of the following: # ''Sk'' × ''Sn−k'' the stabilizer of a ''k''-set (that is, intransitive) # ''Sa '' wr'' Sb'' with ''n'' = ''ab,'' the stabilizer of a partition into ''b'' parts of size ''a'' (that is, imprimitive) #'' primitive'' (that is, preserves no nontrivial partition) and of one of the following types: ::* AGL(''d'',''p'') ::*'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Group
In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to itself). The group of ''all'' permutations of a set ''M'' is the symmetric group of ''M'', often written as Sym(''M''). The term ''permutation group'' thus means a subgroup of the symmetric group. If then Sym(''M'') is usually denoted by S''n'', and may be called the ''symmetric group on n letters''. By Cayley's theorem, every group is isomorphic to some permutation group. The way in which the elements of a permutation group permute the elements of the set is called its group action. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry. Basic properties and terminology Being a subgroup of a symmetric group, all that is necessary for a set of permutatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Simple Group
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album ''Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb A nonfinite verb is a derivative form of a verb unlike finite verbs. Accordingly, nonfinite verb forms are inflected for neither number nor person, and they cannot perform action as the root ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Subgroup
In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup ''H'' of a group ''G'' is a proper subgroup, such that no proper subgroup ''K'' contains ''H'' strictly. In other words, ''H'' is a maximal element of the partially ordered set of subgroups of ''G'' that are not equal to ''G''. Maximal subgroups are of interest because of their direct connection with primitive permutation representations of ''G''. They are also much studied for the purposes of finite group theory: see for example Frattini subgroup, the intersection of the maximal subgroups. In semigroup theory, a maximal subgroup of a semigroup ''S'' is a subgroup (that is, a subsemigroup which forms a group under the semigroup operation) of ''S'' which is not properly contained in another subgroup of ''S''. Notice that, here, there is no requirement that a maximal subgroup be proper, so if ''S'' is in fact a group then its uniq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael O'Nan
Michael Ernest O'Nan (August 9, 1943, Fort Knox, Kentucky – July 31, 2017, Princeton, New Jersey) was an American mathematician, specializing in group theory. O'Nan received his PhD in 1970 from Princeton University under Daniel Gorenstein with thesis ''A Characterization of the Three-Dimensional Projective Unitary Group over a Finite Field''. He was a professor at Rutgers University. In 1976 he found strong evidence for the existence of a sporadic group, which Charles Sims constructed. The group is now named the O'Nan group after O'Nan. The O'Nan–Scott theorem in group theory is also named after O'Nan, who discovered it independently from Leonard Scott. It describes the maximal subgroup In mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra. In group theory, a maximal subgroup ''H'' of a group ''G'' is a proper subgroup, such that no proper subgroup ''K'' contains ''H'' s ...s of the symmetric groups. Selected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Aschbacher
Michael George Aschbacher (born April 8, 1944) is an American mathematician best known for his work on finite groups. He was a leading figure in the completion of the classification of finite simple groups in the 1970s and 1980s. It later turned out that the classification was incomplete, because the case of quasithin groups had not been finished. This gap was fixed by Aschbacher and Stephen D. Smith in 2004, in a pair of books comprising about 1300 pages. Aschbacher is currently the Shaler Arthur Hanisch Professor of Mathematics at the California Institute of Technology. Education and career Aschbacher received his B.S. at the California Institute of Technology in 1966 and his Ph.D. at the University of Wisconsin–Madison in 1969. He joined the faculty of the California Institute of Technology in 1970 and became a full professor in 1976. He was a visiting scholar at the Institute for Advanced Study in 1978–79. He was awarded the Cole Prize in 1980, and was elected to the Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wreath Product
In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used in the classification of permutation groups and also provide a way of constructing interesting examples of groups. Given two groups A and H (sometimes known as the ''bottom'' and ''top''), there exist two variations of the wreath product: the unrestricted wreath product A \text H and the restricted wreath product A \text H. The general form, denoted by A \text_ H or A \text_ H respectively, requires that H acts on some set \Omega; when unspecified, usually \Omega = H (a regular wreath product), though a different \Omega is sometimes implied. The two variations coincide when A, H, and \Omega are all finite. Either variation is also denoted as A \wr H (with \wr for the LaTeX symbol) or ''A'' ≀ ''H'' (Unicode U+2240). The notion ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Of A Set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. Definition and Notation A partition of a set ''X'' is a set of non-empty subsets of ''X'' such that every element ''x'' in ''X'' is in exactly one of these subsets (i.e., ''X'' is a disjoint union of the subsets). Equivalently, a family of sets ''P'' is a partition of ''X'' if and only if all of the following conditions hold: *The family ''P'' does not contain the empty set (that is \emptyset \notin P). *The union of the sets in ''P'' is equal to ''X'' (that is \textstyle\bigcup_ A = X). The sets in ''P'' are said to exhaust or cover ''X''. See also collectively exhaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Permutation Group
In mathematics, a permutation group ''G'' acting on a non-empty finite set ''X'' is called primitive if ''G'' acts transitively on ''X'' and the only partitions the ''G''-action preserves are the trivial partitions into either a single set or into , ''X'', singleton sets. Otherwise, if ''G'' is transitive and ''G'' does preserve a nontrivial partition, ''G'' is called imprimitive. While primitive permutation groups are transitive, not all transitive permutation groups are primitive. The simplest example is the Klein four-group acting on the vertices of a square, which preserves the partition into diagonals. On the other hand, if a permutation group preserves only trivial partitions, it is transitive, except in the case of the trivial group acting on a 2-element set. This is because for a non-transitive action, either the orbits of ''G'' form a nontrivial partition preserved by ''G'', or the group action is trivial, in which case ''all'' nontrivial partitions of ''X'' (which exis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Simple Group
In mathematics, a group is said to be almost simple if it contains a non- abelian simple group and is contained within the automorphism group of that simple group – that is, if it fits between a (non-abelian) simple group and its automorphism group. In symbols, a group ''A'' is almost simple if there is a (non-abelian) simple group ''S'' such that S \leq A \leq \operatorname(S). Examples * Trivially, non-abelian simple groups and the full group of automorphisms are almost simple, but proper examples exist, meaning almost simple groups that are neither simple nor the full automorphism group. * For n=5 or n \geq 7, the symmetric group \mathrm_n is the automorphism group of the simple alternating group \mathrm_n, so \mathrm_n is almost simple in this trivial sense. * For n=6 there is a proper example, as \mathrm_6 sits properly between the simple \mathrm_6 and \operatorname(\mathrm_6), due to the exceptional outer automorphism of \mathrm_6. Two other groups, the Mathieu group \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57â ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Cameron (mathematician)
Peter Jephson Cameron FRSE (born 23 January 1947) is an Australian mathematician who works in group theory, combinatorics, coding theory, and model theory. He is currently half-time Professor of Mathematics at the University of St Andrews, and Emeritus Professor at Queen Mary University of London. Cameron received a B.Sc. from the University of Queensland and a D.Phil. in 1971 from the University of Oxford as a Rhodes Scholar, with Peter M. Neumann as his supervisor. Subsequently, he was a Junior Research Fellow and later a Tutorial Fellow at Merton College, Oxford, and also lecturer at Bedford College, London. Work Cameron specialises in algebra and combinatorics; he has written books about combinatorics, algebra, permutation groups In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
