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Zeros of Brauer characters. (English) Zbl 1274.20007
Summary: The authors obtain a sufficient condition to determine whether an element is a vanishing regular element of some Brauer characters. More precisely, let $$G$$ be a finite group and $$p$$ be a fixed prime, and $$H=G'O^{p'}(G)$$; if $$g\in G^0-H^0$$ with $$o(gH)$$ coprime to the number of irreducible $$p$$-Brauer characters of $$G$$, then there always exists a nonlinear irreducible $$p$$-Brauer character which vanishes on $$g$$. The authors also show in this note that the sums of certain irreducible $$p$$-Brauer characters take the value zero on every element of $$G^0-H^0$$.

##### MSC:
 20C20 Modular representations and characters
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