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Generalized Fitting Subgroup
In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup ''F'' of a finite group ''G'', named after Hans Fitting, is the unique largest normal nilpotent subgroup of ''G''. Intuitively, it represents the smallest subgroup which "controls" the structure of ''G'' when ''G'' is solvable. When ''G'' is not solvable, a similar role is played by the generalized Fitting subgroup ''F*'', which is generated by the Fitting subgroup and the components of ''G''. For an arbitrary (not necessarily finite) group ''G'', the Fitting subgroup is defined to be the subgroup generated by the nilpotent normal subgroups of ''G''. For infinite groups, the Fitting subgroup is not always nilpotent. The remainder of this article deals exclusively with finite groups. The Fitting subgroup The nilpotency of the Fitting subgroup of a finite group is guaranteed by Fitting's theorem which says that the product of a finite collection of normal nilpotent subgroups of ''G'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Chief Series
In abstract algebra, a chief series is a maximal normal series for a group. It is similar to a composition series, though the two concepts are distinct in general: a chief series is a maximal ''normal'' series, while a composition series is a maximal '' subnormal'' series. Chief series can be thought of as breaking the group down into less complicated pieces, which may be used to characterize various qualities of the group. Definition A chief series is a maximal normal series for a group. Equivalently, a chief series is a composition series of the group ''G'' under the action of inner automorphisms. In detail, if ''G'' is a group, then a chief series of ''G'' is a finite collection of normal subgroups ''N''''i'' ⊆ ''G'', :1=N_0\subseteq N_1\subseteq N_2\subseteq\cdots\subseteq N_n=G, such that each quotient group ''N''''i''+1/''N''''i'', for ''i'' = 1, 2,..., ''n'' − 1, is a minimal normal subgroup of ''G''/''N''''i''. Equivalently, there does n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signalizer Functor
In mathematics, in the area of abstract algebra, a signalizer functor is a mapping from a potential finite subgroup to the centralizers of the nontrivial elements of an abelian group. The signalizer functor theorem provides the conditions under which the source of such a functor is in fact a subgroup. The signalizer functor was first defined by Daniel Gorenstein. George Glauberman proved the Solvable Signalizer Functor Theorem for solvable groups and Patrick McBride proved it for general groups. Results concerning signalizer functors play a major role in the classification of finite simple groups. Definition Let ''A'' be a non- cyclic elementary abelian ''p''-subgroup of the finite group ''G.'' An A-signalizer functor on ''G'' (or simply a signalizer functor when ''A'' and ''G'' are clear) is a mapping ''θ'' from the set of nonidentity elements of ''A'' to the set of ''A''-invariant ''p′''-subgroups of ''G'' satisfying the following properties: *For every nonidentity eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bender's Method
In group theory, Bender's method is a method introduced by for simplifying the local group theoretic analysis of the odd order theorem. Shortly afterwards he used it to simplify the Walter theorem on groups with abelian Sylow 2-subgroups , and Gorenstein and Walter's classification of groups with dihedral Sylow 2-subgroups. Bender's method involves studying a maximal subgroup ''M'' containing the centralizer of an involution, and its generalized Fitting subgroup ''F''*(''M''). One succinct version of Bender's method is the result that if ''M'', ''N'' are two distinct maximal subgroups of a simple group with ''F''*(''M'') ≤ ''N'' and ''F''*(''N'') ≤ ''M'', then there is a prime ''p'' such that both ''F''*(''M'') and ''F''*(''N'') are ''p''-groups. This situation occurs whenever ''M'' and ''N'' are distinct maximal parabolic subgroups of a simple group of Lie type, and in this case ''p'' is the characteristic, but this has only been used to help identify groups of low Lie ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classification Of Finite Simple Groups
In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating groups, alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic groups, sporadic (the Tits group is sometimes regarded as a sporadic group because it is not strictly a group of Lie type, in which case there would be 27 sporadic groups). The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Simple groups can be seen as the basic building blocks of all finite groups, reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Of Lie Type
In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a Reductive group, reductive linear algebraic group with values in a finite field. The phrase ''group of Lie type'' does not have a widely accepted precise definition, but the important collection of finite Simple group, simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups. The name "groups of Lie type" is due to the close relationship with the (infinite) Lie groups, since a compact Lie group may be viewed as the rational points of a reductive linear algebraic group over the field of real numbers. and are standard references for groups of Lie type. Classical groups An initial approach to this question was the definition and detailed study of the so-called ''classical groups'' over finite and other field (mathematics), fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Analysis
In algebraic geometry and related areas of mathematics, local analysis is the practice of looking at a problem relative to each prime number ''p'' first, and then later trying to integrate the information gained at each prime into a 'global' picture. These are forms of the localization approach. Group theory In group theory, local analysis was started by the Sylow theorems, which contain significant information about the structure of a finite group ''G'' for each prime number ''p'' dividing the order of ''G''. This area of study was enormously developed in the quest for the classification of finite simple groups, starting with the Feit–Thompson theorem that groups of odd order are solvable. Number theory In number theory one may study a Diophantine equation, for example, modulo ''p'' for all primes ''p'', looking for constraints on solutions. The next step is to look modulo prime powers, and then for solutions in the ''p''-adic field. This kind of local analysis provides c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, generated by a single element. That is, it is a set (mathematics), set of Inverse element, 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 Exponentiation, power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a ''Generating set of a group, generator'' of the group. Every infinite cyclic group is isomorphic to the additive group \Z, the integers. Every finite cyclic group of Order (group theory), order n is isomorphic to the additive group of Quotient group, Z/''n''Z, the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, every characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center of a group. Definition A subgroup of a group is called a characteristic subgroup if for every automorphism of , one has ; then write . It would be equivalent to require the stronger condition = for every automorphism of , because implies the reverse inclusion . Basic properties Given , every automorphism of induces an automorphism of the quotient group , which yields a homomorphism . If has a unique subgroup of a given index, then is characteristic in . Related concepts Normal subgroup A subgroup of that is invariant under all inner automorphisms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Group
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The date of incorporation is listed as 1999 by Companies House of Gibraltar, who class it as a holding company A holding company is a company whose primary business is holding a controlling interest in the Security (finance), securities of other companies. A holding company usually does not produce goods or services itself. Its purpose is to own Share ...; however it is understood that SIMPLE Group's business and trading activities date to the second part of the 90s, probably as an incorporated body. SIMPLE Group Limited is a conglomerate that cultivate secrecy, they are not listed on any Stock Exchange and the group is owned by a complicated series of offshore trusts. The Sunday Times stated that SIMPLE Group's interests could be eva ... [...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/\iota(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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Group
In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients. Examples The smallest (non-trivial) perfect group is the alternating group ''A''5. More generally, any non-abelian simple group is perfect since the commutator subgroup is a normal subgroup with abelian quotient. However, a perfect group need not be simple; for example, the special linear group over the field with 5 elements, SL(2,5) (or the binary icosahedral group, which is isomorphic to it) is perfect but not simple (it has a non-trivial center containing -\!\left(\begin1 & 0 \\ 0 & 1\end\right) = \left(\begin4 & 0 \\ 0 & 4\end\right)). The direct product of any two simple non-abelian groups is perfect but not simple; the commutator of two elements is ''a'',''b''),(''c'',''d'')= ( 'a'',''c'' 'b'',''d''. Since commutators in each simple group form a generating set, pairs of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
