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Supersoluble Group
In mathematics, a group is supersolvable (or supersoluble) if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability. Definition Let ''G'' be a group. ''G'' is supersolvable if there exists a normal series :\ = H_0 \triangleleft H_1 \triangleleft \cdots \triangleleft H_ \triangleleft H_s = G such that each quotient group H_/H_i \; is cyclic and each H_i is normal in G. By contrast, for a solvable group the definition requires each quotient to be abelian. In another direction, a polycyclic group must have a subnormal series with each quotient cyclic, but there is no requirement that each H_i be normal in G. As every finite solvable group is polycyclic, this can be seen as one of the key differences between the definitions. For a concrete example, the alternating group on four points, A_4, is solvable but not supersolvable. Basic Properties Some facts about supersolvable groups: * Supersolvable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Hall Subgroup
In mathematics, specifically group theory, a Hall subgroup of a finite group ''G'' is a subgroup whose order is coprime to its index. They were introduced by the group theorist . Definitions A Hall divisor (also called a unitary divisor) of an integer ''n'' is a divisor ''d'' of ''n'' such that ''d'' and ''n''/''d'' are coprime. The easiest way to find the Hall divisors is to write the prime power factorization of the number in question and take any subset of the factors. For example, to find the Hall divisors of 60, its prime power factorization is 22 × 3 × 5, so one takes any product of 3, 22 = 4, and 5. Thus, the Hall divisors of 60 are 1, 3, 4, 5, 12, 15, 20, and 60. A Hall subgroup of ''G'' is a subgroup whose order is a Hall divisor of the order of ''G''. In other words, it is a subgroup whose order is coprime to its index. If ''π'' is a set of primes, then a Hall ''π''-subgroup is a subgroup whose order is a product of pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Baum's Theorem
eBaum's World is an entertainment website owned by Literally Media. The site was founded in 2001 and features comedy content such as memes, viral videos, images, and other forms of Internet culture. Content is primarily user submitted in exchange for points through a monetary point system "eBones." History of ownership eBaum's World originated in Rochester, New York, featuring entertainment media such as videos, Adobe Flash cartoons, and web games. The site was created and owned by Eric "eBaum" Bauman and his father, Neil. In August 2007, eBaum's World was acquired by HandHeld Entertainment, also known as ZVUE Corporation, for $15 million up front, $2.5 million in HandHeld stock, and up to $52.5 million in cash and stock over 3 years. On January 31, 2009, Bauman and the company's staff were terminated by ZVUE and the company moved to the San Francisco ZVUE offices. As of 2016, the Israel-based company Literally Media held a controlling stake of eBaum's World. eBaum's Worl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Supersolvable Lattice
In mathematics, a supersolvable lattice is a graded poset, graded Lattice (order), lattice that has a maximal total order#Chains, chain of elements, each of which obeys a certain modularity relationship. The definition encapsulates many of the nice properties of lattice of subgroups, lattices of subgroups of supersolvable groups. Motivation A finite group (mathematics), group G is said to be ''supersolvable'' if it admits a maximal subgroup series, chain (or ''series'') of subgroups so that each subgroup in the chain is normal in G. A normal subgroup has been known since the 1940s to be left and (dual) right Modular pair, modular as an element of the lattice of subgroups. Richard P. Stanley, Richard Stanley noticed in the 1970s that certain geometric lattices, such as the partition lattice, obeyed similar properties, and gave a lattice-theoretic abstraction. Definition A finite graded lattice L is supersolvable if it admits a maximal chain \mathbf of elements (called an M-chain o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lattice Of Subgroups
In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial ordering being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection. Example The dihedral group Dih4 has ten subgroups, counting itself and the trivial subgroup. Five of the eight group elements generate subgroups of order two, and the other two non- identity elements both generate the same cyclic subgroup of order four. In addition, there are two subgroups of the form Z2 × Z2, generated by pairs of elements. The lattice formed by these ten subgroups is shown in the illustration. This example also shows that the lattice of all subgroups of a group is not a modular lattice in general. Indeed, this particular lattice contains the forbidden "pentagon" N5 as a sublattice. Properties For any ''A'', ''B'', and ''C'' subgroups of a group with '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
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] [Amazon] |
Monomial Group
In mathematics, in the area of algebra studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters are all monomial, that is, induced from characters of degree 1. In this section only finite groups are considered. A monomial group is solvable. Every supersolvable group and every solvable A-group is a monomial group. Factor groups of monomial groups are monomial, but subgroups need not be, since every finite solvable group can be embedded in a monomial group.As shown by and in textbook form in . The symmetric group S_4 is an example of a monomial group that is neither supersolvable nor an A-group. The special linear group In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix ... \operatorname_2(\mathbb F_3) is the sm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Representation Theory Of Finite Groups
The representation theory of groups is a part of mathematics which examines how groups act on given structures. Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are also considered. For more details, please refer to the section on permutation representations. Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces over fields of characteristic zero. Because the theory of algebraically closed fields of characteristic zero is complete, a theory valid for a special algebraically closed field of characteristic zero is also valid for every other algebraically closed field of characteristic zero. Thus, without loss of generality, we can study vector spaces over \Complex. Representation theory is used in many parts of mathematics, as well as in quantum chemistry and physics. Among other things it is used in algebra to e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Z-group
In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct types of group (mathematics), groups: * in the study of finite groups, a Z-group is a finite group whose Sylow subgroups are all cyclic group, cyclic. * in the study of infinite groups, a Z-group is a group which possesses a very general form of central series. * in the study of linearly ordered group, ordered groups, a Z-group or \mathbb Z-group is a discretely ordered abelian group whose quotient over its minimal convex subgroup is divisible. Such groups are elementary equivalence, elementarily equivalent to the integers (\mathbb Z,+,<). Z-groups are an alternative presentation of Presburger arithmetic. * occasionally, (Z)-group is used to mean a Zassenhaus group, a special type of permutation group. Groups whose Sylow subgroups are cyclic :''Usage: , , , , '' In the study of finite groups, a Z-group is a finite group whose Sylow subgr ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Square-free Integer
In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are Square-free factorization Every positive integer n can be factored in a unique way as n=\prod_^k q_i^i, where the q_i different from one are square-free integers that are pairwise coprime. This is called the ''square-free factorization'' of . To construct the square-free factorization, let n=\prod_^h p_j^ be the prime factorization of n, where the p_j are distinct prime numbers. Then the factors of the square-free factorization are defined as q_i=\prod_p_j. An integer is square-free if and only if q_i=1 for all i > 1. An integer greater than one is the kth power of another integer if and only if k is a divisor of all i such that q_i\neq 1. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
