Richard Brauer
Encyclopedia
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
.
, 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
.
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
and American mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He worked mainly in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, but made important contributions to number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. He was the founder of modular representation theory
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
.
Mathematical work
Several theorems bear his name, including Brauer's induction theoremBrauer'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 finite group...
, which has applications in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
as well as finite group theory, and its corollary Brauer's characterization of characters
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 finite group...
, 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
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
, 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
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
to obtain subtle information about group characters, particularly via his three main theorems
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 with those of its p-local subgroups, that is to say, the normalizers of its non-trivial p-subgroups....
. 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
Quasidihedral group
In mathematics, the quasi-dihedral groups and semi-dihedral groups are non-abelian groups of order a power of 2. For every positive integer n greater than or equal to 4, there are exactly four isomorphism classes of nonabelian groups of order 2n which have a cyclic subgroup of index 2...
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
Gorenstein–Walter theorem
In mathematics, the Gorenstein–Walter theorem, proved by , states that if a finite group G has a dihedral Sylow 2-subgroup, and O is the maximal normal subgroup of odd order, then G/O is isomorphic to a 2-group, or the alternating group A7, or a subgroup of PΓL22 containing PSL2 for q an odd prime...
, classifying finite groups with a dihedral
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
Sylow 2-subgroup, and Glauberman's
George Glauberman
George Glauberman is a mathematician at the University of Chicago who works on finite simple groups. He proved the ZJ theorem and the Z* theorem....
Z* theorem. The theory of a block
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
with a cyclic defect group
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
, first worked out by Brauer in the case when the principal block
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
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
Brauer tree
In mathematical finite group theory, a Brauer tree is a planar tree that encodes the characters of a block with cyclic defect group of a finite group G. In fact, the trees encode the group algebra up to Morita equivalence. Such algebras coming from Brauer trees are called Brauer tree algebras....
is a combinatorial object associated to a block
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
with cyclic defect group which encodes much information about the structure of the block.
In 1970, he was awarded the National Medal of Science
National Medal of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social sciences, biology, chemistry, engineering, mathematics and...
.
See also
- Brauer algebraBrauer algebraIn mathematics, a Brauer algebra is an algebra introduced by used in the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the general linear group in Schur–Weyl duality....
- Brauer groupBrauer groupIn mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
, the equivalence classes of brauer algebras over the same fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
F equipped with a group operation - Brauer–Cartan–Hua theoremBrauer–Cartan–Hua theoremThe Brauer–Cartan–Hua theorem is a theorem in abstract algebra pertaining to division rings, which says that given two division rings K ≤ D such that xKx−1 is contained in K for every x not equal to 0 in D, then either K is contained in Z, the center of D, or...
- Brauer–Nesbitt theorem
- Brauer–Manin obstruction
- Brauer–Siegel theorem
- Brauer–Suzuki theoremBrauer–Suzuki theoremIn mathematics, the Brauer–Suzuki theorem, proved by , , , states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group has a centre of order 2. In particular, such a group cannot be simple.A generalization of the...
- Brauer's theorem
- Brauer's theorem on induced charactersBrauer's theorem on induced charactersBrauer'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 finite group...
- Brauer charactersModular representation theoryModular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...