Richard Brauer

Richard Brauer

Infobox Scientist
name = Richard Brauer
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image_width = 150px
caption = Richard Brauer
birth_date = February 10, 1901
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death_date = April 17, 1977
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nationality = United States, Germany
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field = Scientist, Mathematician
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doctoral_advisor = Issai Schur, Erhard Schmidt
doctoral_students = Cecil J. Nesbitt, Robert Steinberg
known_for = Brauer's theorem on induced characters
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Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation theory.

Several theorems bear his name, including Brauer's induction theorem, which has applications in number theory as well as finite group theory, and its corollary Brauer's characterization of characters, which is central to the theory of group characters.

The Brauer-Fowler theorem, published in 1956, later provided significant impetus towards the classification of finite simple groups, for it implied that there could only be finitely many finite simple groups for which the centralizer of an involution (element of order 2) had a specified structure.

Brauer applied modular representation theory to obtain subtle information about group characters, particularly via his three main theorems. These methods were particularly useful in the classification of finite simple groups with low rank Sylow 2-subgroups. The Brauer-Suzuki theorem showed that no finite simple group could have a generalized quaternion Sylow 2-subgroup, and the Alperin-Brauer-Gorenstein theorem classified finite groups with wreathed or quasidihedral Sylow 2-subgroups. The methods developed by Brauer were also instrumental in contributions by others to the classification program: for example, the Gorenstein-Walter theorem, classifying finite groups with a dihedral Sylow 2-subgroup, and Glauberman's Z* theorem. The theory of a block with a cyclic defect group, first worked out by Brauer in the case when the principal block has defect group of order "p", and later worked out in full generality by E.C. Dade, also had several applications to group theory, for example to finite groups of matrices over the complex numbers in small dimension. The Brauer tree is a combinatorial object associated to a block with cyclic defect group which encodes much information about the structure of the block.

See also

* Brauer algebra, also called central simple algebra
* Brauer group, the equivalence classes of brauer algebras over the same field "F" equipped with a group operation
* Brauer–Nesbitt theorem
* Brauer-Manin obstruction
* Brauer-Siegel theorem
* Brauer's theorem
* Brauer's theorem on induced characters
* Brauer characters



External links

*MacTutor Biography|id=Brauer

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  • Richard Brauer — mit seiner Frau Ilse Brauer, 1970 Richard Dagobert Brauer (* 10. Februar 1901 in Charlottenburg; † 17. April 1977 in Belmont, Massachusetts) war ein deutsch amerikanischer Mathematiker. Inhaltsverzeichn …   Deutsch Wikipedia

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  • Brauer's three main theorems — Brauer s main theorems are three theorems in representation theory of finite groups linking the blocks of a finite group (in characteristic p ) with those of its p local subgroups, that is to say, the normalizers of its non trivial p subgroups.… …   Wikipedia

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  • Brauer's theorem on induced characters — Brauer s theorem on induced characters, often known as Brauer s induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, which is, in turn, part of the representation theory of a …   Wikipedia

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