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Complex Reflection Group
In mathematics, a complex reflection group is a finite group acting on a finite-dimensional 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 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'' \operatorname(r) := \operatorname(r-\operatorname_V) has codimension 1. A (finite) complex reflectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...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 cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''. The index is denoted , G:H, or :H/math> or (G:H). Because ''G'' is the disjoint union of the left cosets and because each left coset has the same size as ''H'', the index is related to the 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 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'' is finite, the formula may be written as , G:H, = , G, /, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Icosahedral Group
In mathematics, the binary icosahedral group 2''I'' or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 120. It is an extension of the icosahedral group ''I'' or (2,3,5) of order 60 by the cyclic group of order 2, and is the preimage of the icosahedral group under the 2:1 covering homomorphism :\operatorname(3) \to \operatorname(3)\, of the special orthogonal group by the spin group. It follows that the binary icosahedral group is a discrete subgroup of Spin(3) of order 120. It should not be confused with the full icosahedral group, which is a different group of order 120, and is rather a subgroup of the orthogonal group O(3). The binary icosahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism \operatorname(3) \cong \operatorname(1) where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Octahedral Group
In mathematics, the binary octahedral group, name as 2O or Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 48. It is an extension of the chiral octahedral group ''O'' or (2,3,4) of order 24 by a cyclic group of order 2, and is the preimage of the octahedral group under the 2:1 covering homomorphism \operatorname(3) \to \operatorname(3) of the special orthogonal group by the spin group. It follows that the binary octahedral group is a discrete subgroup of Spin(3) of order 48. The binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism \operatorname(3) \cong \operatorname(1) where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.) Elements Explicitly, the binary octahedral group is given as the union of the 24 Hurwitz units :\ with al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Tetrahedral Group
In mathematics, the binary tetrahedral group, denoted 2T or , Coxeter&Moser: Generators and Relations for discrete groups: : Rl = Sm = Tn = RST is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of order 12 by a cyclic group of order 2, and is the preimage of the tetrahedral group under the 2:1 covering homomorphism Spin(3) → SO(3) of the special orthogonal group by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The complex reflection group named 3(24)3 by G.C. Shephard or 3 and by Coxeter, is isomorphic to the binary tetrahedral group. The binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism , where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.) Elements E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a '' generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedral Group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, or refers to the symmetries of the -gon, a group of order . In abstract algebra, refers to this same dihedral group. This article uses the geometric convention, . Definition Elements A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. Usually, we take n \ge 3 here. The associated rotations and reflections make up the dihedral group \mathrm_n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides and n/2 axes of symmetry connecting ... [...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 repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extra Special Group
In group theory, a branch of abstract algebra, extraspecial groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime ''p'' and positive integer ''n'' there are exactly two (up to isomorphism) extraspecial groups of order ''p''1+2''n''. Extraspecial groups often occur in centralizers of involutions. The ordinary character theory of extraspecial groups is well understood. Definition Recall that a finite group is called a ''p''-group if its order is a power of a prime ''p''. A ''p''-group ''G'' is called extraspecial if its center ''Z'' is cyclic of order ''p'', and the quotient ''G''/''Z'' is a non-trivial elementary abelian ''p''-group. Extraspecial groups of order ''p''1+2''n'' are often denoted by the symbol ''p''1+2''n''. For example, 21+24 stands for an extraspecial group of order 225. Classification Every extraspecial ''p''-group has order ''p''1+2''n'' for some positive integer ''n'', and conversely for each such number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Product
In mathematics, especially in the field of group theory, the central product is one way of producing a group from two smaller groups. The central product is similar to the direct product, but in the central product two isomorphic central subgroups of the smaller groups are merged into a single central subgroup of the product. Central products are an important construction and can be used for instance to classify extraspecial groups. Definition There are several related but distinct notions of central product. Similarly to the direct product, there are both internal and external characterizations, and additionally there are variations on how strictly the intersection of the factors is controlled. A group ''G'' is an internal central product of two subgroups ''H'', ''K'' if # ''G'' is generated by ''H'' and ''K''. # Every element of ''H'' commutes with every element of ''K''. Sometimes the stricter requirement that H\cap K is exactly equal to the center is imposed, as in . The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a '' generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperoctahedral Group
In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a parameter , the dimension of the hypercube. As a Coxeter group it is of type , and as a Weyl group it is associated to the symplectic groups and with the orthogonal groups in odd dimensions. As a wreath product it is S_2 \wr S_n where is the symmetric group of degree . As a permutation group, the group is the signed symmetric group of permutations ''π'' either of the set or of the set such that for all . As a matrix group, it can be described as the group of orthogonal matrices whose entries are all integers. Equivalently, this is the set of matrices with entries only 0, 1, or –1, which are invertible, and which have exactly one non-zero entry in each row or column. The representation theory of the hyperoctahedral grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
