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Covering Groups Of The Alternating And Symmetric Groups
In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations of the alternating and symmetric groups. The covering groups were classified in : for , the covering groups are 2-fold covers except for the alternating groups of degree 6 and 7 where the covers are 6-fold. For example the binary icosahedral group covers the icosahedral group, an alternating group of degree 5, and the binary tetrahedral group covers the tetrahedral group, an alternating group of degree 4. Definition and classification A group homomorphism from ''D'' to ''G'' is said to be a Schur cover of the finite group ''G'' if: # the kernel is contained both in the center and the commutator subgroup of ''D'', and # amongst all such homomorphisms, this ''D'' has maximal size. The Schur multiplier of ''G'' is the kernel of any Schur cover and has many interpretations. When the homomorphism is understood, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Linear Group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of invertible matrices of real numbers, and is denoted by GL''n''(R) or . More generally, the general linear group of degree ''n'' over any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sporadic Group
In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group, in which case there would be 27 sporadic groups. The monster group is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it. Names Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Extension (mathematics)
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overset\;G\;\overset\;Q \to 1. If G is an extension of Q by N, then G is a group, \iota(N) is a normal subgroup of G and the quotient group G/\iota(N) is isomorphic to the group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined. Note that the phrasing "G is an extension of N by Q" is also used by some. Since any finite group G possesses a maximal normal subgroup N with simple factor group G/N, all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup N lies in the cente ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overset\;G\;\overset\;Q \to 1. If G is an extension of Q by N, then G is a group, \iota(N) is a normal subgroup of G and the quotient group G/\iota(N) is isomorphic to the group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined. Note that the phrasing "G is an extension of N by Q" is also used by some. Since any finite group G possesses a maximal normal subgroup N with simple factor group G/N, all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup N lies in the center o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pin Group
The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies. History and shareholding The PIN Group originally traded under the name ''Briefdienstleistungsunternehmen PIN intelligente Dienstleistungen AG'', a postal services company founded in 1999 in Berlin. In 2004, Axel Springer AG and the Holtzbrinck Publishing Group each bought a majority shareholding of 30% from the company founders and venture capital company DKB Wagniskapital GmbH. In October 2005 the remaining shares of the PIN founders were acquired by WAZ-Mediengruppe and the Luxembourg investment company Rosalia Investment S.A. before founding the PIN Group. In the summer of 2006 the newspaper publishers Madsack, M. DuMont Schauberg, Rheinisch-Bergische Verlagsgesellschaft and W. Girardet merged their postal services divisions into the company, receiving in return a 10 per cent holding of PIN Gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a Lie group, Spin(''n'') therefore shares its dimension, , and its Lie algebra with the special orthogonal group. For , Spin(''n'') is simply connected and so coincides with the universal cover of SO(''n''). The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −. Spin(''n'') can be constructed as a subgroup of the invertible elements in the Clifford algebra Cl(''n''). A distinct article discusses the spin representations. Motivation and physical interpretation The spin group is used in physics to describe the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used to describe electrical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Group
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension ''d'' is then a subgroup of the orthogonal group O(''d''). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules. Each point group can be represented as sets of orthogonal matrices ''M'' that transform point ''x'' into point ''y'' according to Each element of a point group is either a rotation (determinant of ''M'' = 1), or it is a reflection or improper rotation (determinant of ''M'' = −1). The geometric symmetries of crystals are described by space groups, which allow translations and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a ''k''-simplex is a ''k''-dimensional polytope which is the convex hull of its ''k'' + 1 vertices. More formally, suppose the ''k'' + 1 points u_0, \dots, u_k \in \mathbb^ are affinely independent, which means u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points : C = \left\ This representation in terms of weighted vertices is known as the barycentric coordinate system. A regular simplex is a simplex that is also a regular polytope. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Representation
In the mathematical field of representation theory, a trivial representation is a representation of a group ''G'' on which all elements of ''G'' act as the identity mapping of ''V''. A trivial representation of an associative or Lie algebra is a ( Lie) algebra representation for which all elements of the algebra act as the zero linear map (endomorphism) which sends every element of ''V'' to the zero vector. For any group or Lie algebra, an irreducible trivial representation always exists over any field, and is one-dimensional, hence unique up to isomorphism. The same is true for associative algebras unless one restricts attention to unital algebras and unital representations. Although the trivial representation is constructed in such a way as to make its properties seem tautologous, it is a fundamental object of the theory. A subrepresentation is equivalent to a trivial representation, for example, if it consists of invariant vectors; so that searching for such subrepresentation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Matrices
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when used to multiply another matrix, say , results in permuting the rows (when pre-multiplying, to form ) or columns (when post-multiplying, to form ) of the matrix . Definition Given a permutation of ''m'' elements, :\pi : \lbrace 1, \ldots, m \rbrace \to \lbrace 1, \ldots, m \rbrace represented in two-line form by :\begin 1 & 2 & \cdots & m \\ \pi(1) & \pi(2) & \cdots & \pi(m) \end, there are two natural ways to associate the permutation with a permutation matrix; namely, starting with the ''m'' × ''m'' identity matrix, , either permute the columns or permute the rows, according to . Both methods of defining permutation matrices appear in the literature and the properties expressed in one representation can be easily converted to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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