Zeros of Primitive Characters

Wenyang Wangand Ni Du

1Center for General Education,Xiamen Huaxia University,Xiamen 361024,China.

2School of Mathematical Sciences,Xiamen University,Xiamen 361005,China.

Abstract.Let G be a finite group. An irreducible character χ of G is said to be primitive if χϑG for any characterϑ of a proper subgroup of G.In this paper,we consider about the zeros of primitive characters.Denote by Irrpri(G)the set of all irreducible primitive characters of G.We proved that if g∈G and the order of gG′in the factor group G/G′does not divide|Irrpri(G)|,then there exists ϕ ∈ Irrpri(G)such that ϕ(g)=0.

Key words:Finite group,primitive character,vanishing element.

1 Introduction

LetGbe a finite group and Irr(G)be the set of all irreducible characters ofG.For an elementgofG,gis called a vanishing element if there existsχ∈ Irr(G)such thatχ(g)=0.In[3],W.Burnside proved that for any nonlinear irreducible characterχ,there always existsg∈Gsuch thatχ(g)=0,which means that there exists at least a vanishing element for any nonlinear irreducible characterχ.It is interesting to investigate when an element of a finite group can be a vanishing element.In[1],G.Chen obtained a sufficient condition to determine when an element is a vanishing element.More precisely,suppose thatg∈G−G′and the order ofgG′in the factor groupG/G′is coprime to|Irr(G)|,then there existsχ∈ Irr(G)such thatχ(g)=0.In[4],H.Wang,X.Chen and J.Zeng showed a similar sufficient condition about the Brauer characters.In[2],X.Chen and G.Chen investigated the monomial Brauer characters.An irreducible characterχofGis said to be primitive ifχϑGfor any characterϑof a proper subgroup ofG.In this paper,we consider about the zeros of primitive characters.

2 Main results and proofs

Acknowledgments

The project was supported by the Natural Science Foundation of China(Grant No.11771356),the Natural Science Foundation of Fujian Province of China(No.2019J01025)and the Research Fund for Fujian Young Faculty(Grant No.JAT190985).