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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''''p''′(''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''''p''′ is the maximal normal subgroup of ''G'' of order coprime to ''p'', the ''p''′-core *''O''''p'' is the maximal normal ''p''-subgroup of ''G'', the ''p''-core. *''O''''p''′,''p''(''G'') is the maximal normal ''p''-nilpotent subgroup of ''G'', the ''p''′,''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 p-subgroup of ''G'' such that ''POp′''(''G'') is nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Glauberman
George Isaac Glauberman (born 1941) is a mathematician at the University of Chicago who works on finite simple groups. He proved the ZJ theorem and the Z* theorem. Born in New York City on March 3, 1941, Glauberman did his undergraduate studies at the Polytechnic Institute of Brooklyn, graduating in 1961, and earned a master's degree from Harvard University in 1962. He obtained his PhD degree from the University of Wisconsin–Madison in 1965, under the supervision of Richard Bruck. He has had 22 PhD students, including Ahmed Chalabi and Peter Landrock, the president and founder of Cryptomathic. He has co-authored with J. L. Alperin, Simon P. Norton, Zvi Arad, and Justin Lynd. In 1970 he was an invited speaker at the International Congress of Mathematicians at Nice. In 2012 he became a fellow of the American Mathematical Society. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-group
In mathematics, specifically group theory, given a prime number ''p'', a ''p''-group is a group in which the order of every element is a power of ''p''. That is, for each element ''g'' of a ''p''-group ''G'', there exists a nonnegative integer ''n'' such that the product of ''pn'' copies of ''g'', and not fewer, is equal to the identity element. The orders of different elements may be different powers of ''p''. Abelian ''p''-groups are also called ''p''-primary or simply primary. A finite group is a ''p''-group if and only if its order (the number of its elements) is a power of ''p''. Given a finite group ''G'', the Sylow theorems guarantee the existence of a subgroup of ''G'' of order ''pn'' for every prime power ''pn'' that divides the order of ''G''. Every finite ''p''-group is nilpotent. The remainder of this article deals with finite ''p''-groups. For an example of an infinite abelian ''p''-group, see Prüfer group, and for an example of an infinite simple ''p''-grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canadian Journal Of Mathematics
The ''Canadian Journal of Mathematics'' (french: Journal canadien de mathématiques) is a bimonthly mathematics journal published by the Canadian Mathematical Society. It was established in 1949 by H. S. M. Coxeter and G. de B. Robinson. The current editors-in-chief of the journal are Louigi Addario-Berry and Eyal Goren. The journal publishes articles in all areas of mathematics. See also * Canadian Mathematical Bulletin The ''Canadian Mathematical Bulletin'' (french: Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antoni ... References External links * University of Toronto Press academic journals Mathematics journals Publications established in 1949 Bimonthly journals Multilingual journals Cambridge University Press academic journals Academic journals associated with learned and professional societies of Canada ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centralizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', or equivalently, such that conjugation by g leaves each element of ''S'' fixed. The normalizer of ''S'' in ''G'' is the set of elements \mathrm_G(S) of ''G'' that satisfy the weaker condition of leaving the set S \subseteq G fixed under conjugation. The centralizer and normalizer of ''S'' are subgroups of ''G''. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets ''S''. Suitably formulated, the definitions also apply to semigroups. In ring theory, the centralizer of a subset of a ring is defined with respect to the semigroup (multiplication) operation of the ring. The centralizer of a subset of a ring ''R'' is a subring of ''R''. This article also deals with centralizers and normaliz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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''-stable group *The 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'p''′(''G'')''Z''(''J''(''S'')) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper P-series
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. Significance Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup of any point acts as the identity on its entire orbit. Thus, in ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P′,p-core
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. Significance Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup of any point acts as the identity on its entire orbit. Thus, in ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-nilpotent Group
In mathematical group theory, a normal p-complement of a finite group for a prime ''p'' is a normal subgroup of order coprime to ''p'' and index a power of ''p''. In other words the group is a semidirect product of the normal ''p''-complement and any Sylow ''p''-subgroup. A group is called p-nilpotent if it has a normal ''p''-complement. Cayley normal 2-complement theorem Cayley showed that if the Sylow 2-subgroup of a group ''G'' is cyclic then the group has a normal 2-complement, which shows that the Sylow 2-subgroup of a simple group of even order cannot be cyclic. Burnside normal p-complement theorem showed that if a Sylow ''p''-subgroup of a group ''G'' is in the center of its normalizer then ''G'' has a normal ''p''-complement. This implies that if ''p'' is the smallest prime dividing the order of a group ''G'' and the Sylow ''p''-subgroup is cyclic, then ''G'' has a normal ''p''-complement. Frobenius normal p-complement theorem The Frobenius normal ''p''-complement the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-core
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. Significance Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup of any point acts as the identity on its entire orbit. Thus, in ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P′-core
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. Significance Normal cores are important in the context of group actions on sets, where the normal core of the isotropy subgroup of any point acts as the identity on its entire orbit. Thus, in ca ... [...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] |
Center (group Theory)
In abstract algebra, the center of a group, , is the set of elements that commute with every element of . It is denoted , from German '' Zentrum,'' meaning ''center''. In set-builder notation, :. The center is a normal subgroup, . As a subgroup, it is always characteristic, but is not necessarily fully characteristic. The quotient group, , is isomorphic to the inner automorphism group, . A group is abelian if and only if . At the other extreme, a group is said to be centerless if is trivial; i.e., consists only of the identity element. The elements of the center are sometimes called central. As a subgroup The center of ''G'' is always a subgroup of . In particular: # contains the identity element of , because it commutes with every element of , by definition: , where is the identity; # If and are in , then so is , by associativity: for each ; i.e., is closed; # If is in , then so is as, for all in , commutes with : . Furthermore, the center of is always ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |