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Wolfgang Gaschütz
Wolfgang Gaschütz (11 June 1920 – 7 November 2016) was a German mathematician, known for his research in group theory, especially the theory of finite groups. (article written by L. A. Shemetkov & R. Schmidt) Biography Gaschütz was born on 11 June 1920 in Karlshof, Oderbruch. He moved with his family in 1931 to Berlin, where he completed his ''Abitur'' in 1938. He served as an artillery officer in WW II, which ended for him in 1945 near Kiel. There in autumn 1945 he matriculated at the University of Kiel. He was inspired by Andreas Speiser's book ''Die Theorie der Gruppen von endlicher Ordnung''. Gaschütz received his Ph.D. ( ''Promotion'') in 1949 under the supervision of Karl-Heinrich Weise with doctoral dissertation entitled ''(Zur \Phi-Untergruppe endlicher Gruppen)''. In 1953 Gaschütz completed his habilitation in Kiel. At the University of Kiel he held the junior academic appointments ''Wissenschaftliche Hilfskraft'' from 1949 to 1956 and ''Diätendozent'' from 1956 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Warwick
The University of Warwick ( ; abbreviated as ''Warw.'' in post-nominal letters) is a public research university on the outskirts of Coventry between the West Midlands and Warwickshire, England. The university was founded in 1965 as part of a government initiative to expand higher education. The Warwick Business School was established in 1967, the Warwick Law School in 1968, Warwick Manufacturing Group (WMG) in 1980, and Warwick Medical School in 2000. Warwick incorporated Coventry College of Education in 1979 and Horticulture Research International in 2004. Warwick is primarily based on a campus on the outskirts of Coventry, with a satellite campus in Wellesbourne and a central London base at the Shard. It is organised into three faculties—Arts; Science, Engineering and Medicine, and Social Sciences—within which there are thirty-two departments. Warwick has around 29,534 full-time students and 2,691 academic and research staff, with an average intake of 4,950 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carter Subgroup
In mathematics, especially in the field of group theory, a Carter subgroup of a finite group ''G'' is a self-normalizing subgroup of ''G'' that is nilpotent. These subgroups were introduced by Roger Carter, and marked the beginning of the post 1960 theory of solvable groups . proved that any finite solvable group has a Carter subgroup, and all its Carter subgroups are conjugate subgroups (and therefore isomorphic). If a group is not solvable it need not have any Carter subgroups: for example, the alternating group A5 of order 60 has no Carter subgroups. showed that even if a finite group is not solvable then any two Carter subgroups are conjugate. A Carter subgroup is a maximal nilpotent subgroup, because of the normalizer condition for nilpotent groups, but not all maximal nilpotent subgroups are Carter subgroups . For example, any non-identity proper subgroup of the nonabelian group of order six is a maximal nilpotent subgroup, but only those of order two are Carter subgr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall Subgroup
In mathematics, specifically group theory, a Hall subgroup of a finite group ''G'' is a subgroup whose order is coprime to its index. They were introduced by the group theorist . Definitions A Hall divisor (also called a unitary divisor) of an integer ''n'' is a divisor ''d'' of ''n'' such that ''d'' and ''n''/''d'' are coprime. The easiest way to find the Hall divisors is to write the prime power factorization of the number in question and take any subset of the factors. For example, to find the Hall divisors of 60, its prime power factorization is 22 × 3 × 5, so one takes any product of 3, 22 = 4, and 5. Thus, the Hall divisors of 60 are 1, 3, 4, 5, 12, 15, 20, and 60. A Hall subgroup of ''G'' is a subgroup whose order is a Hall divisor of the order of ''G''. In other words, it is a subgroup whose order is coprime to its index. If ''π'' is a set of primes, then a Hall ''π''-subgroup is a subgroup whose order is a product of pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Zeitschrift
''Mathematische Zeitschrift'' ( German for ''Mathematical Journal'') is a mathematical journal for pure and applied mathematics published by Springer Verlag. History The journal was founded in 1917, with its first issue appearing in 1918. It was initially edited by Leon Lichtenstein together with Konrad Knopp, Erhard Schmidt, and Issai Schur. Because Lichtenstein was Jewish, he was forced to step down as editor in 1933 under the Nazi rule of Germany; he fled to Poland and died soon after. The editorship was offered to Helmut Hasse Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ..., but he refused, Translated by Bärbel Deninger from the 1982 German original. and Konrad Knopp took it over. Other past editors include Erich Kamke, Friedrich Karl Schmidt, Rolf Nevanlinna, Hel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formation (group Theory)
In group theory, a branch of mathematics, a formation is a class of groups closed under taking images and such that if ''G''/''M'' and ''G''/''N'' are in the formation then so is ''G''/''M''∩''N''. introduced formations to unify the theory of Hall subgroups and Carter subgroups of finite solvable groups. Some examples of formations are the formation of ''p''-groups for a prime ''p'', the formation of π-groups for a set of primes π, and the formation of nilpotent groups. Special cases A Melnikov formation is closed under taking quotients, normal subgroups and group extensions. Thus a Melnikov formation ''M'' has the property that for every short exact sequence :1 \rightarrow A \rightarrow B \rightarrow C \rightarrow 1\ ''A'' and ''C'' are in ''M'' if and only if ''B'' is in ''M''. A full formation is a Melnikov formation which is also closed under taking subgroups. An almost full formation is one which is closed under quotients, direct products and subgroups, but not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philip Hall
Philip Hall FRS (11 April 1904 – 30 December 1982), was an English mathematician. His major work was on group theory, notably on finite groups and solvable groups. Biography He was educated first at Christ's Hospital, where he won the Thompson Gold Medal for mathematics, and later at King's College, Cambridge. He was elected a Fellow of the Royal Society in 1951 and awarded its Sylvester Medal in 1961. He was President of the London Mathematical Society from 1955–1957, and was awarded its Berwick Prize in 1958 and De Morgan Medal in 1965. Publications * * * See also * Abstract clone * Commutator collecting process * Isoclinism of groups * Regular p-group * Three subgroups lemma * Hall algebra, and Hall polynomials * Hall subgroup In mathematics, specifically group theory, a Hall subgroup of a finite group ''G'' is a subgroup whose order is coprime to its index. They were introduced by the group theorist . Definitions A Hall divisor (also called a unitary ... [...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 equations. 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 The smallest Galois field extension of \mathbb containing the elementa = \sqr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group ''G'' in an associated ''G''-module ''M'' to elucidate the properties of the group. By treating the ''G''-module as a kind of topological space with elements of G^n representing ''n''- simplices, topological properties of the space may be computed, such as the set of cohomology groups H^n(G,M). The cohomology groups in turn provide insight into the structure of the group ''G'' and ''G''-module ''M'' themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frattini Subgroup
In mathematics, particularly in group theory, the Frattini subgroup \Phi(G) of a group is the intersection of all maximal subgroups of . For the case that has no maximal subgroups, for example the trivial group or a Prüfer group, it is defined by \Phi(G)=G. It is analogous to the Jacobson radical in the theory of rings, and intuitively can be thought of as the subgroup of "small elements" (see the "non-generator" characterization below). It is named after Giovanni Frattini, who defined the concept in a paper published in 1885. Some facts * \Phi(G) is equal to the set of all non-generators or non-generating elements of . A non-generating element of is an element that can always be removed from a generating set; that is, an element ''a'' of such that whenever is a generating set of containing ''a'', X \setminus \ is also a generating set of . * \Phi(G) is always a characteristic subgroup of ; in particular, it is always a normal subgroup of . * If is finite, then \P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Wielandt
__NOTOC__ Helmut Wielandt (19 December 1910 – 14 February 2001) was a German mathematician who worked on permutation groups. He was born in Niedereggenen, Lörrach, Germany. He gave a plenary lecture ''Entwicklungslinien in der Strukturtheorie der endlichen Gruppen'' (Lines of Development in the Structure Theory of Finite Groups) at the International Congress of Mathematicians (ICM) in 1958 at Edinburgh and was an Invited Speaker with talk ''Bedingungen für die Konjugiertheit von Untergruppen endlicher Gruppen'' (Conditions for the Conjugacy of Finite Groups) at the ICM in 1962 in Stockholm. Among his work in Algebra is an elegant proof of the Sylow Theorems (replacing an older cumbersome proof involving double cosets) that is in the standard textbooks on Abstract Algebra, i.e. Group Theory. See also * Collatz–Wielandt formula * Wielandt theorem In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers z for which \mathrm\,z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Steinitz
Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician. Biography Steinitz was born in Laurahütte ( Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and his wife Auguste Cohen; he had two brothers. He studied at the University of Breslau and the University of Berlin, receiving his Ph.D. from Breslau in 1894. Subsequently, he took positions at Charlottenburg (now Technische Universität Berlin), Breslau, and the University of Kiel, Germany, where he died in 1928. Steinitz married Martha Steinitz and had one son. Mathematical works Steinitz's 1894 thesis was on the subject of projective configurations; it contained the result that any abstract description of an incidence structure of three lines per point and three points per line could be realized as a configuration of straight lines in the Euclidean plane with the possible exception of one of the lines. His thesis also contains th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
