Zeros of Monomial Brauer Characters

Citation:

Xiaoyou CHEN,Gang CHEN.Zeros of Monomial Brauer Characters[J].Chinese Annals of Mathematics B,2019,40(2):213~216
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Authors:

Xiaoyou CHEN; Gang CHEN

Foundation:

This work was supported by the National Natural Science Foundation of China (Nos.11571129, 11771356), the Natural Key Fund of Education Department of Henan Province (No.17A110004) and the Natural Funds of Henan Province (Nos.182102410049, 162300410066).
Abstract: Let $G$ be a finite group and $p$ be a fixed prime. A $p$-Brauer character of $G$ is said to be monomial if it is induced from a linear $p$-Brauer character of some subgroup (not necessarily proper) of $G$. Denote by ${\rm IBr}_{m}(G)$ the set of irreducible monomial $p$-Brauer characters of $G$. Let $H=G'{\bf O}^{p'}(G)$ be the smallest normal subgroup such that $G/H$ is an abelian $p'$-group. Suppose that $g\in G$ is a $p$-regular element and the order of $gH$ in the factor group $G/H$ does not divide $|{\rm IBr}_{m}(G)|$. Then there exists $\varphi\in {\rm IBr}_{m}(G)$ such that $\varphi(g)=0$.

Keywords:

Brauer character, Finite group, Vanishing regular element, Monomial Brauer character

Classification:

20C15, 20C20
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