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Exceptional Character
In mathematical finite group theory, an exceptional character of a group is a character related in a certain way to a character of a subgroup. They were introduced by , based on ideas due to Brauer in . Definition Suppose that ''H'' is a subgroup of a finite group ''G'', and ''C''1, ..., ''C''''r'' are some conjugacy classes of ''H'', and φ1, ..., φ''s'' are some irreducible characters of ''H''. Suppose also that they satisfy the following conditions: #''s'' ≥ 2 #φ''i'' = φ''j'' outside the classes ''C''1, ..., ''C''''r'' #φ''i'' vanishes on any element of ''H'' that is conjugate in ''G'' but not in ''H'' to an element of one of the classes ''C''1, ..., ''C''''r'' #If elements of two classes are conjugate in ''G'' then they are conjugate in ''H'' #The centralizer in ''G'' of any element of one of the classes ''C''1,...,''C''''r'' is contained in ''H'' Then ''G'' has ''s'' irreducible characters ''s''1,...,''s''''s'', ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite 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 is a verb that is not capable of serving as the main verb in an independent clause * a non-finite clause In linguistics, a non-finite clause is a dependent or embedded clause that represen ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character (mathematics)
In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified. Multiplicative character A multiplicative character (or linear character, or simply character) on a group ''G'' is a group homomorphism from ''G'' to the multiplicative group of a field , usually the field of complex numbers. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of ''G''. Sometimes only ''unitary'' characters are considered (thus the image is in the unit circle); other such homomorphisms are then called ''quasi-characters''. Dirichlet characters can be seen as a special case of this definition. Multiplicative characters are linearly independent, i.e. if \chi_1,\chi_2, \ldots , \chi_n are different cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dade Isometry
In mathematical finite group theory, the Dade isometry is an isometry from class function on a subgroup ''H'' with support on a subset ''K'' of ''H'' to class functions on a group ''G'' . It was introduced by as a generalization and simplification of an isometry used by in their proof of the odd order theorem, and was used by in his revision of the character theory of the odd order theorem. Definitions Suppose that ''H'' is a subgroup of a finite group ''G'', ''K'' is an invariant subset of ''H'' such that if two elements in ''K'' are conjugate in ''G'', then they are conjugate in ''H'', and π a set of primes containing all prime divisors of the orders of elements of ''K''. The Dade lifting is a linear map ''f'' → ''f''σ from class functions ''f'' of ''H'' with support on ''K'' to class functions ''f''σ of ''G'', which is defined as follows: ''f''σ(''x'') is ''f''(''k'') if there is an element ''k'' ∈ ''K'' conjugate to the π-part of ''x'', and 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Set Of Characters
In mathematical representation theory, coherence is a property of sets of characters that allows one to extend an isometry from the degree-zero subspace of a space of characters to the whole space. The general notion of coherence was developed by , as a generalization of the proof by Frobenius of the existence of a Frobenius kernel of a Frobenius group and of the work of Brauer and Suzuki on exceptional characters. developed coherence further in the proof of the Feit–Thompson theorem that all groups of odd order are solvable. Definition Suppose that ''H'' is a subgroup of a finite group ''G'', and ''S'' a set of irreducible characters of ''H''. Write ''I''(''S'') for the set of integral linear combinations of ''S'', and ''I''0(''S'') for the subset of degree 0 elements of ''I''(''S''). Suppose that τ is an isometry from ''I''0(''S'') to the degree 0 virtual characters of ''G''. Then τ is called coherent if it can be extended to an isometry from ''I''(''S'') to characters of ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and public-domain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original pagination and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Groups
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 is a verb that is not capable of serving as the main verb in an independent clause * a non-finite clause In linguistics, a non-finite clause is a dependent or embedded clause that represen ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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