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In mathematics, the 'Brauer-Nesbitt theorem' can refer to several different theorems proved by Richard Brauer and 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.
★ 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.
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.
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| References |
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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