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Mitchell Group
In mathematics, Mitchell's group is a complex reflection group in 6 complex dimensions of order 108 × 9!, introduced by . It has the structure 6.PSU4(F3).2. As a complex reflection group it has 126 reflections of order 2, and its ring of invariants is a polynomial algebra with generators of degrees 6, 12, 18, 24, 30, 42. Coxeter gives it group symbol 2 3sup>3 and Coxeter-Dynkin diagram .Coxeter, ''Finite Groups Generated by Unitary Reflections'', 1966, 4. ''The Graphical Notation'', Table of n-dimensional groups generated by n Unitary Reflections. pp. 422–423 Mitchell's group is an index 2 subgroup of the automorphism group of the Coxeter–Todd lattice In mathematics, the Coxeter–Todd lattice K12, discovered by , is a 12-dimensional even integral lattice of discriminant 36 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 3, and is a .... References Further reading * * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Reflection Group
In mathematics, a complex reflection group is a Group (mathematics), finite group acting on a finite-dimensional vector space, finite-dimensional complex numbers, complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise. Complex reflection groups arise in the study of the invariant theory of polynomial rings. In the mid-20th century, they were completely classified in work of Shephard and Todd. Special cases include the symmetric group of permutations, the dihedral groups, and more generally all finite real reflection groups (the Coxeter groups or Weyl groups, including the symmetry groups of regular polyhedron, regular polyhedra). Definition A (complex) reflection ''r'' (sometimes also called ''pseudo reflection'' or ''unitary reflection'') of a finite-dimensional complex vector space ''V'' is an element r \in GL(V) of finite order that fixes a complex hyperplane pointwise, that is, the ''fixed-space'' \operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Algebra
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (mathematics), ring formed from the set (mathematics), set of polynomials in one or more indeterminate (variable), indeterminates (traditionally also called variable (mathematics), variables) with coefficients in another ring (mathematics), ring, often a field (mathematics), field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the Integer#Algebraic_properties, integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, formal power series, rings of formal power series, Ore polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Set Of A Group
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group (mathematics), group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their Inverse element, inverses. In other words, if S is a subset of a group G, then \langle S\rangle, the ''subgroup generated by S'', is the smallest subgroup of G containing every element of S, which is equal to the intersection over all subgroups containing the elements of S; equivalently, \langle S\rangle is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.) If G=\langle S\rangle, then we say that S ''generates'' G, and the elements in S are called ''generators'' or ''group generators''. If S is the empty set, then \langle S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University. He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada. He was an author of 12 books, including ''The Fifty-Nine Icosahedra'' (1938) and ''Regular Polytopes'' (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. Biography Coxeter was born in Kensington, England, to Harold Samuel Coxeter an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Index Of A Subgroup
In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the number of left Coset, cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''. The index is denoted , G:H, or [G:H] or (G:H). Because ''G'' is the disjoint union of the left cosets and because each left coset has the same cardinality, size as ''H'', the index is related to the order (group theory), orders of the two groups by the formula :, G, = , G:H, , H, (interpret the quantities as cardinal numbers if some of them are infinite). Thus the index , G:H, measures the "relative sizes" of ''G'' and ''H''. For example, let G = \Z be the group of integers under addition, and let H = 2\Z be the subgroup consisting of the Parity (mathematics), even integers. Then 2\Z has two cosets in \Z, namely the set of even integers and the set of odd integers, so the index , \Z:2\Z, is 2. More generally, , \Z:n\Z, = n for any positive integer ''n''. When ''G'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗. More precisely, is a subgroup of if the Restriction (mathematics), restriction of ∗ to is a group operation on . This is often denoted , read as " is a subgroup of ". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group is a subgroup which is a subset, proper subset of (that is, ). This is often represented notationally by , read as " is a proper subgroup of ". Some authors also exclude the trivial group from being proper (that is, ). If is a subgroup of , then is sometimes called an overgroup of . The same definitions apply more generally when is an arbitrary se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism Group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the group of invertible linear transformations from ''X'' to itself (the general linear group of ''X''). If instead ''X'' is a group, then its automorphism group \operatorname(X) is the group consisting of all group automorphisms of ''X''. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. Examples If ''X'' is a set with no additional structure, then any bijection from ''X'' to itself is an automorphism, and hence the automorphism group of ''X'' in this case is precisely the symmetric group of ''X''. If the set ''X'' has additional structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter–Todd Lattice
In mathematics, the Coxeter–Todd lattice K12, discovered by , is a 12-dimensional even integral lattice of discriminant 36 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 3, and is analogous to the Barnes–Wall lattice. The automorphism group of the Coxeter–Todd lattice has order 210·37·5·7=78382080, and there are 756 vectors in this lattice of norm 4 (the shortest nonzero vectors in this lattice). Properties The Coxeter–Todd lattice can be made into a 6-dimensional lattice self dual over the Eisenstein integers. The automorphism group of this complex lattice has index 2 in the full automorphism group of the Coxeter–Todd lattice and is a complex reflection group (number 34 on the list) with structure 6.PSU4(F3).2, called the Mitchell group. The genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant tax ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proceedings Of The Cambridge Philosophical Society
''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure and applied mathematics. The journal, titled ''Proceedings of the Cambridge Philosophical Society'' before 1975, has been published since 1843. Abstracting and indexing The journal is abstracted and indexed in *MathSciNet *Science Citation Index Expanded *Scopus *ZbMATH Open See also *Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ... External linksofficial website References Academic journals associated with learned and professional societies Cambridge University Press academic journals Mathematics e ... [...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 (mathematician), 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, Hen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
