HOME





P-constrained Group
In mathematics, a p-constrained group is a finite group resembling the centralizer of an element of prime order ''p'' in a group of Lie type over a finite field of characteristic ''p''. They were introduced by in order to extend some of Thompson's results about odd groups to groups with dihedral Sylow 2-subgroups. Definition If a group has trivial ''p'' core Core or cores may refer to: Science and technology * Core (anatomy), everything except the appendages * Core (laboratory), a highly specialized shared research resource * Core (manufacturing), used in casting and molding * Core (optical fiber ... O''p''(''G''), then it is defined to be ''p''-constrained if the ''p''-core O''p''(''G'') contains its centralizer, or in other words if its generalized Fitting subgroup is a ''p''-group. More generally, if O''p''(''G'') is non-trivial, then ''G'' is called ''p''-constrained if ''G''/O''p''(''G'') is . All ''p''-solvable groups are ''p''-constrained. See also * ''p''- ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

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]  


Finite Group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be bu ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ...
[...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]  


picture info

Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod n, integers mod p when p is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set that is a fiel ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Characteristic (algebra)
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said to have characteristic zero. That is, is the smallest positive number such that: : \underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: : \underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). This definition applies in the more general class of rngs (see '); for (unital) rings the two definitions are equivalent due to their distributive law. ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Dihedral Group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, 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 n-gon, -gon, a group of order . In abstract algebra, refers to this same dihedral group. This article uses the geometric convention, . Definition The word "dihedral" comes from "di-" and "-hedron". The latter comes from the Greek word hédra, which means "face of a geometrical solid". Overall it thus refers to the two faces of a polygon. Elements A regular polygon with n sides has 2n different symmetries: n rotational symmetry, rotational symmetries and n reflection symmetry, reflection symmetries. Usually, we take n ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  




Core (group Theory)
In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the ''p''-core of a group. The normal core Definition For a group ''G'', the normal core or normal interiorRobinson (1996) p.16 of a subgroup ''H'' is the largest normal subgroup of ''G'' that is contained in ''H'' (or equivalently, the intersection of the conjugates of ''H''). More generally, the core of ''H'' with respect to a subset ''S'' ⊆ ''G'' is the intersection of the conjugates of ''H'' under ''S'', i.e. :\mathrm_S(H) := \bigcap_. Under this more general definition, the normal core is the core with respect to ''S'' = ''G''. The normal core of any normal subgroup is the subgroup itself. Dual to the concept of normal core is that of which is the smallest normal subgroup of ''G'' containing ''H''. Significance Normal cores are important in the context of group actions on sets ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


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]  


P-solvable Group
In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the ''p''-core of a group. The normal core Definition For a group ''G'', the normal core or normal interiorRobinson (1996) p.16 of a subgroup ''H'' is the largest normal subgroup of ''G'' that is contained in ''H'' (or equivalently, the intersection of the conjugates of ''H''). More generally, the core of ''H'' with respect to a subset ''S'' ⊆ ''G'' is the intersection of the conjugates of ''H'' under ''S'', i.e. :\mathrm_S(H) := \bigcap_. Under this more general definition, the normal core is the core with respect to ''S'' = ''G''. The normal core of any normal subgroup is the subgroup itself. Dual to the concept of normal core is that of which is the smallest normal subgroup of ''G'' containing ''H''. Significance Normal cores are important in the context of group actions on sets, ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


P-stable Group
In finite group theory, a ''p''-stable group for an odd prime ''p'' is a finite group satisfying a technical condition introduced by in order to extend Thompson's uniqueness results in the odd order theorem to groups with dihedral Sylow 2-subgroups. Definitions There are several equivalent definitions of a ''p''-stable group. ;First definition. We give definition of a ''p''-stable group in two parts. The definition used here comes from . 1. Let ''p'' be an odd prime and ''G'' be a finite group with a nontrivial ''p''-core O_p(G). Then ''G'' is ''p''-stable if it satisfies the following condition: Let ''P'' be an arbitrary ''p''-subgroup of ''G'' such that O_(G) is a normal subgroup of ''G''. Suppose that x \in N_G(P) and \bar x is the coset of C_G(P) containing ''x''. If ,x,x1, then \overline\in O_n(N_G(P)/C_G(P)). Now, define \mathcal_p(G) as the set of all ''p''-subgroups of ''G'' maximal with respect to the property that O_p(M)\not= 1. 2. Let ''G'' be a finite gr ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  




ZJ Theorem
In mathematics, George Glauberman's ZJ theorem states that if a finite group ''G'' is ''p''-constrained and ''p''-stable and has a normal ''p''-subgroup for some odd prime ''p'', then ''O'''(''G'')''Z''(''J''(''S'')) is a normal subgroup of ''G'', for any Sylow ''p''-subgroup ''S''. Notation and definitions *''J''(''S'') is the Thompson subgroup of a ''p''-group ''S'': the subgroup generated by the abelian subgroups of maximal order. *''Z''(''H'') means the center of a group ''H''. *''O''' is the maximal normal subgroup of ''G'' of order coprime to ''p'', the '-core *''O''''p'' is the maximal normal ''p''-subgroup of ''G'', the ''p''-core. *''O''',''p''(''G'') is the maximal normal ''p''-nilpotent subgroup of ''G'', the ',''p''-core, part of the upper ''p''-series. *For an odd prime ''p'', a group ''G'' with ''O''''p''(''G'') ≠ 1 is said to be ''p''-stable if whenever ''P'' is a of ''G'' such that ''PO''(''G'') is normal in ''G'', and 'P'',''x'',''x''= 1, then the ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]