For the American museum director, see Richard H. W. Brauer.
Not to be confused with L. . E. J. Brouwer
|Born||(1901-02-10)February 10, 1901|
|Died||April 17, 1977(1977-04-17) (aged 76)
Belmont, Massachusetts, U.S.
|Alma mater||University of Berlin (Ph.D., 1926)|
|Known for||Brauer's theorem on induced characters|
|Awards||Cole Prize in Algebra (1949)
National Medal of Science (1970)
|Institutions||University of Toronto, University of Michigan, Harvard University|
|Thesis||Über die Darstellung der Drehungsgruppe durch Gruppen linearer Substitutionen (1926)|
|Doctoral advisor||Issai Schur|
|Doctoral students||R. H. Bruck|
He was the founder of modular representation theory.
Education and career
Alfred Brauer was Richard's brother and seven years older.
They were born to a Jewish family.
Both were interested in science and mathematics, but Alfred was injured in combat in World War I.
He soon transferred to University of Berlin.
on 16 March 1926.
Issai Schur conducted a seminar and posed a problem in 1921 that Alfred and Richard worked on together, and published a result.
The problem also was solved by Heinz Hopf at the same time.
Richard wrote his thesis under Schur, providing an algebraic approach to irreducible, continuous, finite-dimensional representations of real orthogonal (rotation) groups.
Ilse Karger also studied mathematics at the University of Berlin; she and Richard were married 17 September 1925.
Their sons George Ulrich (born 1927) and Fred Gunther (born 1932) also became mathematicians.
When the Nazi Party took over in 1933, the Emergency Committee in Aid of Displaced Foreign Scholars took action to help Brauer and other Jewish scientists.
Brauer was offered an assistant professorship at University of Kentucky.
Richard accepted the offer, and by the end of 1933 he was in Lexington, Kentucky, teaching in English.
Ilse followed the next year with George and Fred; brother Alfred made it to the United States in 1939, but their sister Alice was killed in the Holocaust.
Richard and Nathan Jacobson edited Weyl's lectures Structure and Representation of Continuous Groups.
Brauer also conducted international research with Tadasi Nakayama on representations of algebras.
In 1941 University of Wisconsin hosted visiting professor Brauer.
With his graduate student K. A. Fowler, Brauer proved the Brauer–Fowler theorem.
Donald John Lewis was another of his students at University of Michigan.
In 1952 Brauer joined the faculty of Harvard University.
Before retiring in 1971 he taught aspiring mathematicians such as Donald Passman and I. . Martin Isaacs
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.
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. , also had several applications to group theory, for example to finite groups of matrices over the complex numbers in small dimension. C. Dade
In 1970, he was awarded the National Medal of Science.
Main article: Hypercomplex number
By the 1930s there was evident need to update Study’s article, and Richard Brauer was commissioned to write on the topic for the project.
As it turned out, when Brauer had his manuscript prepared in Toronto in 1936, though it was accepted for publication, politics and war intervened.
It also appeared posthumously as paper #22 in the first volume of his Collected Papers.
His title was "Algebra der hyperkomplexen Zahlensysteme (Algebren)".
Unlike the articles by Study and Cartan, which were exploratory, Brauer’s article reads as a modern abstract algebra text with its universal coverage.
Consider his introduction:
- In the beginning of the 19th century, the usual complex numbers and their introduction through computations with number-pairs or points in the plane, became a general tool of mathematicians. Naturally the question arose whether or not a similar "hypercomplex" number can be defined using points of n-dimensional space. As it turns out, such extension of the system of real numbers requires the concession of some of the usual axioms (Weierstrass 1863). The selection of rules of computation, which cannot be avoided in hypercomplex numbers, naturally allows some choice. Yet in any cases set out, the resulting number systems allow a unique theory with regard to their structural properties and their classification. Further, one desires that these theories stand in close connection with other areas of mathematics, wherewith the possibility of their applications is given.
While still in Königsberg in 1929, Brauer published an article in Mathematische Zeitschrift "Über Systeme hyperkomplexer Zahlen" which was primarily concerned with integral domains (Nullteilerfrei systeme) and the field theory which he used later in Toronto.
- Brauer, R.; Sah, Chih-han, eds. (1969), , W. A. Benjamin, Inc., New York-Amsterdam, MR
- Brauer, R. (1980), Fong, Paul; Wong, Warren J. (eds.), Collected Papers. Vol. I, Mathematicians of Our Time, 17, MIT Press, ISBN 978-0-262-02135-7, MR
- Brauer, R. (1980), Fong, Paul; Wong, Warren J. (eds.), , Mathematicians of Our Time, 18, MIT Press, ISBN 978-0-262-02148-7, MR
- Brauer, R. (1980), Fong, Paul; Wong, Warren J. (eds.), Collected Papers. Vol. III, Mathematicians of Our Time, 19, MIT Press, ISBN 978-0-262-02149-4, MR
- Brauer algebra
- Brauer group, the equivalence classes of brauer algebras over the same field F equipped with a group operation
- Brauer–Cartan–Hua theorem
- Brauer–Nesbitt theorem
- Brauer–Manin obstruction
- Brauer–Siegel theorem
- Brauer–Suzuki theorem
- Brauer's theorem
- Brauer's theorem on induced characters
- Brauer characters
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Richard Brauer.