Brauer–Nesbitt theorem

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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.

[edit] 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.