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Primitive Permutation Representation
In mathematics, a permutation group ''G'' Group action, acting on a non-empty finite set ''X'' is called primitive if ''G'' acts transitive action, transitively on ''X'' and the only Partition_of_a_set, 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 orbit (group theory), orbits of ''G'' form a nontrivial partition preserved by ''G'', or the group action is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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 re ... [...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 Elsevier ( ) is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell (journal), Cell'', the ScienceDirect collection of electronic journals, .... ''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 ( R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colva Roney-Dougal
Colva Mary Roney-Dougal is a British mathematician specializing in group theory and computational algebra. She is Professor of Pure Mathematics at the University of St Andrews, and the Director of the Centre for Interdisciplinary Research in Computational Algebra at St Andrews. She is also known for her popularization of mathematics on BBC radio shows, including appearances on '' In Our Time'' about the mathematics of Emmy Noether and Pierre-Simon Laplace and on '' The Infinite Monkey Cage'' about the nature of infinity and numbers in the real world. Education Roney-Dougal completed her PhD at the University of London in 2001. Her dissertation, ''Permutation Groups with a Unique Non-diagonal Self-paired Orbital'', was supervised by Peter Cameron. Book With John Bray and Derek Holt, Roney-Dougal is the co-author of the book ''The Maximal Subgroups of the Low-Dimensional Finite Classical Groups'' (London Mathematical Society and Cambridge University Press, 2013). Recognition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Jordan's Theorem (symmetric Group)
In finite group theory, Jordan's theorem states that if a primitive permutation group ''G'' is a subgroup of the symmetric group ''S''''n'' and contains a ''p''- cycle for some prime number ''p'' < ''n'' − 2, then ''G'' is either the whole symmetric group ''S''''n'' or the ''A''''n''. It was first proved by . The statement can be generalized to the case that ''p'' is a prime power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number.
For example: , and are prime power ...
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Block (permutation Group Theory)
In mathematics and group theory, a block system for the action of a group ''G'' on a set ''X'' is a partition of ''X'' that is ''G''-invariant. In terms of the associated equivalence relation on ''X'', ''G''-invariance means that :''x'' ~ ''y'' implies ''gx'' ~ ''gy'' for all ''g'' ∈ ''G'' and all ''x'', ''y'' ∈ ''X''. The action of ''G'' on ''X'' induces a natural action of ''G'' on any block system for ''X''. The set of orbits of the ''G''-set ''X'' is an example of a block system. The corresponding equivalence relation is the smallest ''G''-invariant equivalence on ''X'' such that the induced action on the block system is trivial. The partition into singleton sets is a block system and if ''X'' is non-empty then the partition into one set ''X'' itself is a block system as well (if ''X'' is a singleton set then these two partitions are identical). A transitive (and thus non-empty) ''G''-set ''X'' is said to be primitive if it has no other block systems. For a non-empt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
OEIS
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009, and is its chairman. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 370,000 sequences, and is growing by approximately 30 entries per day. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. There is also an advanced search function called SuperSeeker which runs a large number of different algorithms to identify sequences related to the input. History Neil Sloane started coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Carmichael
Robert Daniel Carmichael (March 1, 1879 – May 2, 1967) was an American mathematician. Biography Carmichael was born in Goodwater, Alabama. He attended Lineville College, briefly, and he earned his bachelor's degree in 1898, while he was studying towards his Ph.D. degree at Princeton University. Carmichael completed the requirements for his Ph.D. in mathematics in 1911. Carmichael's Ph.D. research in mathematics was done under the guidance of the noted American mathematician G. David Birkhoff, and it is considered to be the first significant American contribution to the knowledge of differential equations in mathematics. Carmichael next taught at Indiana University from 1911 to 1915. Then he moved on to the University of Illinois, where he remained from 1915 until his retirement in 1947. Carmichael is known for his research in what are now called the Carmichael numbers (a subset of Fermat pseudoprimes, numbers satisfying properties of primes described by Fermat's Littl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Representation
In group theory, the induced representation is a group representation, representation of a group, , which is constructed using a known representation of a subgroup . Given a representation of '','' the induced representation is, in a sense, the "most general" representation of that extends the given one. Since it is often easier to find representations of the smaller group than of '','' the operation of forming induced representations is an important tool to construct new representations''.'' Induced representations were initially defined by Ferdinand Georg Frobenius, Frobenius, for linear representations of finite groups. The idea is by no means limited to the case of finite groups, but the theory in that case is particularly well-behaved. Constructions Algebraic Let be a finite group and any subgroup of . Furthermore let be a representation of . Let be the Index of a subgroup, index of in and let be a full set of representatives in of the Coset, left cosets in . Th ... [...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 uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
