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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 \(G^0-H^0\).

20C20 Modular representations and characters
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