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N-group Theorem
In mathematical finite group theory, an N-group is a group all of whose local subgroups (that is, the normalizers of nontrivial ''p''-subgroups) are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups. Simple N-groups The simple N-groups were classified by in a series of 6 papers totaling about 400 pages. The simple N-groups consist of the special linear groups PSL2(''q''), PSL3(3), the Suzuki groups Sz(22''n''+1), the unitary group U3(3), the alternating group ''A''7, the Mathieu group M11, and the Tits group. (The Tits group was overlooked in Thomson's original announcement in 1968, but Hearn pointed out that it was also a simple N-group.) More generally Thompson showed that any non-solvable N-group is a subgroup of Aut(''G'') containing ''G'' for some simple N-group ''G''. generalized Thompson's theorem to the case of groups where all 2-local subgroups are solvable. The only extra simple groups t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group Theory
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album '' Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb is a verb that is not capable of serving as the main verb in an independent clause * a non-finite clause In linguistics, a non-finite clause is a dependent or embedded clause that represen ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Subgroup
In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). Properties of a point on a function Perhaps the best-known example of the idea of locality lies in the concept of local minimum (or local maximum), which is a point in a function whose functional value is the smallest (resp., largest) within an immediate neighborhood of points. This is to be contrasted with the idea of global minimum (or global maximum), which corresponds to the minimum (resp., maximum) of the function across its entire domain. Properties of a single space A topological space is sometimes said to exhibit a property locally, if the property is exhibited "near" each point in one of the following ways: # Each point has a neighborhood exhibiting the property; # Each point has a neighborhood base of sets exhibiting the propert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solvable Group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Motivation Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0). This means associated to a polynomial f \in F /math> there is a tower of field extensionsF = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m=Ksuch that # F_i = F_ alpha_i/math> where \alpha_i^ \in F_, so \alpha_i is a solution to the equation x^ - a where a \in F_ # F_m contains a splitting field for f(x) Example For example, the smallest Galois field extension of \mathbb containing the elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John G
John is a common English name and surname: * John (given name) * John (surname) John may also refer to: New Testament Works * Gospel of John, a title often shortened to John * First Epistle of John, often shortened to 1 John * Second Epistle of John, often shortened to 2 John * Third Epistle of John, often shortened to 3 John People * John the Baptist (died c. AD 30), regarded as a prophet and the forerunner of Jesus Christ * John the Apostle (lived c. AD 30), one of the twelve apostles of Jesus * John the Evangelist, assigned author of the Fourth Gospel, once identified with the Apostle * John of Patmos, also known as John the Divine or John the Revelator, the author of the Book of Revelation, once identified with the Apostle * John the Presbyter, a figure either identified with or distinguished from the Apostle, the Evangelist and John of Patmos Other people with the given name Religious figures * John, father of Andrew the Apostle and Saint Peter * Pope Joh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Linear Group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant :\det\colon \operatorname(n, F) \to F^\times. where ''F''× is the multiplicative group of ''F'' (that is, ''F'' excluding 0). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When ''F'' is a finite field of order ''q'', the notation is sometimes used. Geometric interpretation The special linear group can be characterized as the group of ''volume and orientation preserving'' linear transformations of R''n''; this corresponds to the interpretation of the determinant as measuring change in volume and orientation. Lie subgroup When ''F'' is R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suzuki Groups
In the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(22''n''+1), 2''B''2(22''n''+1), Suz(22''n''+1), or ''G''(22''n''+1), form an infinite family of groups of Lie type found by , that are simple for ''n'' ≥ 1. These simple groups are the only finite non-abelian ones with orders not divisible by 3. Constructions Suzuki originally constructed the Suzuki groups as subgroups of SL4(F22''n''+1) generated by certain explicit matrices. Ree Ree observed that the Suzuki groups were the fixed points of exceptional automorphisms of some symplectic groups of dimension 4, and used this to construct two further families of simple groups, called the Ree groups. In the lowest case the symplectic group ''B''2(2)≈''S''6; its exceptional automorphism fixes the subgroup Sz(2) or 2''B''2(2), of order 20. gave a detailed exposition of Ree's observation. Tits constructed the Suzuki groups as the symmetries of a certain ovoid in 3-dimensional projective spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternating Group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic properties For , the group A''n'' is the commutator subgroup of the symmetric group S''n'' with index 2 and has therefore ''n''!/2 elements. It is the kernel of the signature group homomorphism explained under symmetric group. The group A''n'' is abelian if and only if and simple if and only if or . A5 is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group. The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto . We have the exact sequence . In Galois theory, this map, or rather the corresponding map , corresponds to associating the Lagrange resolvent cubic to a quartic, which allow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Group
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They were the first sporadic groups to be discovered. Sometimes the notation ''M''9, ''M''10, ''M''20 and ''M''21 is used for related groups (which act on sets of 9, 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones. John Conway has shown that one can also extend this sequence up, obtaining the Mathieu groupoid ''M''13 acting on 13 points. ''M''21 is simple, but is not a sporadic group, being isomorphic to PSL(3,4). History introduced the group ''M''12 as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Group
In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order : 211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a 27th sporadic group. History and properties The Ree groups 2''F''4(22''n''+1) were constructed by , who showed that they are simple if ''n'' ≥ 1. The first member of this series 2''F''4(2) is not simple. It was studied by who showed that it is almost simple, its derived subgroup 2''F''4(2)′ of index 2 being a new simple group, now called the Tits group. The group 2''F''4(2) is a group of Lie type and has a BN pair, but the Tits group itself does not have a BN pair. Because the Tits group is not strictly a group of Lie type, it is sometimes regarded as a 27th sporadic group.For instance, by the ATLAS of Finite Groups and itweb-based descendant/ref> The Schur multiplier of the Tits group is trivial and its outer automorphism group has orde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Abelian
In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian groups are a particular kind of ''p''-group. The case where ''p'' = 2, i.e., an elementary abelian 2-group, is sometimes called a Boolean group. Every elementary abelian ''p''-group is a vector space over the prime field with ''p'' elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/''p''Z)''n'' for ''n'' a non-negative integer (sometimes called the group's ''rank''). Here, Z/''p''Z denotes the cyclic group of order ''p'' (or equivalently the integers mod ''p''), and the superscript notation means the ''n''-fold direct product of groups. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C-group
The C-Group culture is an archaeological culture found in Lower Nubia, which dates from ca. 2400 BCE to ca. 1550 BCE. It was named by George A. Reisner. With no central site and no written evidence about what these people called themselves, Reisner assigned the culture a letter. The C-Group arose after Reisner's A-Group and B-Group cultures, and around the time the Old Kingdom was ending in Ancient Egypt. Overview While today many scholars see A and B as actually being a continuation of the same group, C-Group is considered as the product of Saharan pastoralist distinct. The C-Group is marked by its distinctive pottery, and for its tombs. Early C-Group tombs consisted of a simple " stone circle" with the body buried in a depression in the centre. The tombs later became more elaborate with the bodies being placed in a stone lined chamber, and then the addition of an extra chamber on the east: for offerings. The origins of the C-Group are still debated. Some scholars see it la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Simple Group
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album ''Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb A nonfinite verb is a derivative form of a verb unlike finite verbs. Accordingly, nonfinite verb forms are inflected for neither number nor person, and they cannot perform action as the root ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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