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.
Another version states that if k is a field of characteristic zero, A is a k-algebra, V, W are semisimple A-modules which are finite dimensional over k, and TrV = TrW as elements of Homk(A,k), then V and W are isomorphic as A-modules.
- 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.
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