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Trichotomy Theorem
In group theory, the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by for rank 3 and by for rank at least 4. The three classes are groups of GF(2) type (classified by Timmesfeld and others), groups of "standard type" for some odd prime (classified by the Gilman–Griess theorem In finite group theory, a mathematical discipline, the Gilman–Griess theorem, proved by , classifies the finite simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (al ... and work by several others), and groups of uniqueness type, where Aschbacher proved that there are no simple groups. References * * * Theorems about finite groups {{algebra-stub ... [...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] |
Finite 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 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] |
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; 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 trust An offshore trust is a conventional trust that is formed under the laws of an offshore jurisdiction. Generally offshore trusts are similar in nature and effect to their onshore counterparts; they involve a settlor transferring (or 'settling') a ...s. The Sunday Times stated that SIMPLE Group's interests could be eval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic 2 Type
In finite group theory, a branch of mathematics, a group is said to be of characteristic 2 type or even type or of even characteristic if it resembles a group of Lie type over a field of characteristic 2. In the classification of finite simple groups, there is a major division between group of characteristic 2 type, where involutions resemble unipotent elements, and other groups, where involutions resemble semisimple elements. Groups of characteristic 2 type and rank at least 3 are classified by the trichotomy theorem. Definitions A group is said to be of even characteristic if :C_M(O_2(M)) \le O_2(M) for all maximal 2-local subgroups ''M'' that contain a Sylow 2-subgroup of ''G''. If this condition holds for all maximal 2-local subgroups ''M'' then ''G'' is said to be of characteristic 2 type. use a modified version of this called even type. References * *{{Citation , last1=Gorenstein , first1=D. , author1-link=Daniel Gorenstein , last2=Lyons , first2=Richard , last3=So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank (group Theory)
In the mathematical subject of group theory, the rank of a group ''G'', denoted rank(''G''), can refer to the smallest cardinality of a generating set for ''G'', that is : \operatorname(G)=\min\. If ''G'' is a finitely generated group, then the rank of ''G'' is a nonnegative integer. The notion of rank of a group is a group-theoretic analog of the notion of dimension of a vector space. Indeed, for ''p''-groups, the rank of the group ''P'' is the dimension of the vector space ''P''/Φ(''P''), where Φ(''P'') is the Frattini subgroup. The rank of a group is also often defined in such a way as to ensure subgroups have rank less than or equal to the whole group, which is automatically the case for dimensions of vector spaces, but not for groups such as affine groups. To distinguish these different definitions, one sometimes calls this rank the subgroup rank. Explicitly, the subgroup rank of a group ''G'' is the maximum of the ranks of its subgroups: : \operatorname(G)=\max_ \min\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Groups Of GF(2) Type
In mathematical finite group theory, a group of GF(2)-type is a group with an involution centralizer whose generalized Fitting subgroup is a group of symplectic type . As the name suggests, many of the groups 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 linear algebraic group with values in a finite field. The phra ... over the field with 2 elements are groups of GF(2)-type. Also 16 of the 26 sporadic groups are of GF(2)-type, suggesting that in some sense sporadic groups are somehow related to special properties of the field with 2 elements. showed roughly that groups of GF(2)-type can be subdivided into 8 types. The groups of each of these 8 types were classified by various authors. They consist mainly of groups of Lie type with all roots of the same length over the field with 2 elements, but also include many exceptional cases ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gilman–Griess Theorem
In finite group theory, a mathematical discipline, the Gilman–Griess theorem, proved by , classifies the finite simple groups of characteristic 2 type with ''e''(''G'') ≥ 4 that have a "standard component", which covers one of the three cases of the trichotomy theorem In group theory, the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by for rank 3 and by for rank at least 4. The three classes are groups of GF(2) .... References * Theorems about finite groups {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniqueness Case
In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem In group theory, the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by for rank 3 and by for rank at least 4. The three classes are groups of GF( .... The uniqueness case covers groups ''G'' of characteristic 2 type with ''e''(''G'') ≥ 3 that have an almost strongly ''p''-embedded maximal 2-local subgroup for all primes ''p'' whose 2-local ''p''-rank is sufficiently large (usually at least 3). proved that there are no finite simple groups in the uniqueness case. References * * *{{Citation , last1=Stroth , first1=Gernot , editor1-last=Arasu , editor1-first=K. T. , editor2-last=Dillon , editor2-first=J. F. , editor3-last=Harada , editor3-first=Koichiro , editor4-last=Sehgal , editor4-fir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
