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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 A ''permutation group'' is a subgroup of a symmetric group; that is, its elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Simple Group
In mathematics, a group (mathematics), group is said to be almost simple if it contains a non-abelian group, 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), where the inclusion of S in \mathrm(S) is the Conjugation (group action), action by conjugation, which is Faithful action, faithful since S has a trivial Center (group theory), center. Examples * Trivially, non-abelian simple groups and the full group of automorphisms are almost simple. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Socle (mathematics)
In mathematics, the term socle has several related meanings. Socle of a group In the context of group theory, the socle of a group ''G'', denoted soc(''G''), is the subgroup generated by the minimal normal subgroups of ''G''. It can happen that a group has no minimal non-trivial normal subgroup (that is, every non-trivial normal subgroup properly contains another such subgroup) and in that case the socle is defined to be the subgroup generated by the identity. The socle is a direct product of minimal normal subgroups. As an example, consider the cyclic group Z12 with generator ''u'', which has two minimal normal subgroups, one generated by ''u''4 (which gives a normal subgroup with 3 elements) and the other by ''u''6 (which gives a normal subgroup with 2 elements). Thus the socle of Z12 is the group generated by ''u''4 and ''u''6, which is just the group generated by ''u''2. The socle is a characteristic subgroup, and hence a normal subgroup. It is not necessarily transitively ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object. Definition In an algebraic structure such as a group, a ring, or vector space, an ''automorphism'' is simply a bijective homomorphism of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.) More generally, for an object in some category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism f: X\to X is an automorphism if there is a morphism g: X\to X such that g\circ f= f\circ g = \operatorname _X, where \operatorname _X is the identity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Saxl
Jan Saxl (5 June 1948 – 2 May 2020) was a Czech-British mathematician, and a professor at the University of Cambridge. He was known for his work in finite group theory, particularly on consequences of the classification of finite simple groups. Education and career Saxl was born in Brno, in what was at the time Czechoslovakia. He came to the United Kingdom in 1968, during the Prague Spring. After undergraduate studies at the University of Bristol, he completed his DPhil in 1973 at the University of Oxford under the direction of Peter M. Neumann, with the title of ''Multiply Transitive Permutation Groups''. Saxl held postdoctoral positions at Oxford and the University of Illinois at Chicago, and a lecturer position at the University of Glasgow. He moved to the University of Cambridge in 1976, and spent the rest of his career there. He was elected as a fellow of Gonville and Caius College in 1986, and he retired in 2015. Saxl published around 100 papers, and according to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cheryl Praeger
Cheryl Elisabeth Praeger (born 7 September 1948, Toowoomba, Queensland) is an Australian mathematician. Praeger received BSc (1969) and MSc degrees from the University of Queensland (1974), and a doctorate from the University of Oxford in 1973 under direction of Peter M. Neumann. She has published widely and has advised 27 PhD students (as of March 2018). She is currently Emeritus Professor of Mathematics at the University of Western Australia. She is best known for her works in group theory, algebraic graph theory and combinatorial designs. Education Praeger completed her high school education at Brisbane Girls Grammar School. After graduating high school, Praeger went to the government vocational guidance section to inquire about how she could further study mathematics. The vocational guidance officer she spoke with tried to discourage her from studying mathematics further, suggesting she become a teacher or a nurse because two other girls who came to him wanting to study ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Cameron (mathematician)
Peter Jephson Cameron Fellow of the Royal Society of Edinburgh, FRSE (born 23 January 1947) is an Australian mathematician who works in group theory, combinatorics, coding theory, and model theory. He is currently Emeritus Professor at the University of St Andrews and Queen Mary University of London. Education Cameron received a B.Sc. from the University of Queensland and a D.Phil. in 1971 from the University of Oxford as a Rhodes Scholarship, 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, Bedford College, London. Work Cameron specialises in algebra and combinatorics; he has written books about combinatorics, algebra, permutation groups, and logic, and has produced over 350 academic papers. In 1988, he posed the Cameron–Erdős conjecture with Paul Erdős. Honours and awards He was awarded the London Mathematical Soci ... [...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 57–58 ... [...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 exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Simple Group
In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates. Summary The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families. (In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A8 = ''A''3(2) and ''A''2(4) both have order 20160, and that the group ''Bn''(''q'') has the same order as ''Cn''(''q'') for ''q'' odd, ''n'' > 2. The smalle ... [...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 Empty set, non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a Set (mathematics), 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., the subsets are nonempty mutually disjoint sets). 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 (set theory), union of the sets in ''P'' is equal to ''X'' (that is \textstyle\bigcup_ A = X). The sets in ''P'' are said ... [...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 variants 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 variants coincide when A, H, and \Omega are all finite. Either variant is also denoted as A \wr H (with \wr for the LaTeX symbol) or ''A'' ≀ ''H'' (Unicode U+2240). The notion gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
