Richard Brauer

Richard Brauer
Richard and Ilse Brauer in 1970 Photo courtesy MFO
Born	February 10, 1901 Charlottenburg
Died	April 17, 1977 Belmont, Massachusetts
Nationality	United States, Germany
Fields	Scientist, Mathematician
Doctoral advisor	Issai Schur, Erhard Schmidt
Doctoral students	Cecil J. Nesbitt, Robert Steinberg
Known for	Brauer's theorem on induced characters

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.

Mathematical work

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.

In 1970, he was awarded the National Medal of Science.

See also

• 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

Publications

• Brauer, R.; Sah, Chih-han, eds. (1969), Theory of finite groups: A symposium, W. A. Benjamin, Inc., New York-Amsterdam, MR0240186, http://books.google.com/books?id=YRdCAAAAIAAJ 
• 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, MR581120 
• Brauer, R. (1980), Fong, Paul; Wong, Warren J., eds., Collected papers. Vol. II, Mathematicians of Our Time, 18, MIT Press, ISBN 978-0-262-02148-7, MR581120 
• 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, MR581120 

References

• Curtis, Charles W. (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2677-5, MR1715145, http://books.google.com/books?isbn=0821826778  Review
• Feit, Walter (1979), "Richard D. Brauer", American Mathematical Society. Bulletin. New Series 1 (1): 1–20, doi:10.1090/S0273-0979-1979-14547-6, ISSN 0002-9904, MR513747 

External links

• O'Connor, John J.; Robertson, Edmund F., "Richard Brauer", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Brauer.html .
• Richard Brauer at the Mathematics Genealogy Project.