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# 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.

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 Tr_{V} = Tr_{W} as elements of Hom_{k}(*A*,k), then *V* and *W* are isomorphic as *A*-modules.

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.

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