Brauer–Nesbitt theorem

In mathematics, the Brauer-Nesbitt theorem can refer to several different theorems proved by Richard Brauer and Cecil J. Nesbitt in the representation theory of finite groups.

In modular representation theory,the Brauer-Nesbitt theorem on blocks of defect zero states that a character whose order is divisible by the highest power of a prime "p" dividing the order of a finite group remains irreducible when reduced mod "p" and vanishes on all elements whose order is divisible by "p". Moreover it belongs to a block of defect zero. A block of defect zero contains only one ordinary character and only one modular character.

References

*Curtis, Reiner, "Representation theory of finite groups and associative algebras", Wiley 1962.
*Brauer, R.; Nesbitt, C. "On the modular characters of groups." Ann. of Math. (2) 42, (1941). 556-590.