Theorem of brewers

The set of Brauer is a theorem from the mathematical field of representation theory of groups of Richard Brauer . It says that every linear representation of a finite group can be obtained in a more or less simple way from representations of so-called elementary subgroups. These elementary subgroups are the direct product of a p-group and a cyclic group. To understand the representation theory of , it is sufficient to know the representations of its cyclic and p-subgroups. ${\displaystyle G}$ ${\displaystyle G}$

notation

First we need some definitions:

A group is called -elementary if it is the direct product of a cyclic group of prime order with a -group . ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$

A subgroup of is called elementary if it is -elementary for at least one prime number . ${\displaystyle G}$ ${\displaystyle p}$ ${\displaystyle p}$

A representation of is called monomial if it is induced by a -dimensional representation of a subgroup of . ${\displaystyle G}$ ${\displaystyle 1}$ ${\displaystyle G}$

Theorem of brewers

Each character of a finite group is an integer linear combination of characters that are induced by characters of elementary subgroups . ${\displaystyle G}$

Evidence and a more detailed explanation of Brauer's theory can be found in books by Jean-Pierre Serre and Serge Lang .

Applications

Since -elementary groups are nilpotent and thus solvable , the following theorem can be used: ${\displaystyle p}$

sentence

Be an insoluble group. Then every irreducible representation of is induced by a -dimensional representation of a subgroup of. That is, every irreducible representation of is monomial. ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle 1}$ ${\displaystyle G.}$ ${\displaystyle G}$

With this we get as a consequence from Brauer's theorem:

sentence

Each character of is an integer linear combination of monomial characters. ${\displaystyle G}$

Web links

• The induction theorem from Brauer

Individual evidence

1. ^ Jean-Pierre Serre: Linear Representations of Finite Groups. Springer Verlag, New York 1977, ISBN 0-387-90190-6 .
2. ^ Serge Lang: Algebra. Springer-Verlag, New York 2002, ISBN 0-387-95385-X , pp. 663-729.
3. ↑ Serre, op. Cit.