On Type Of Zeros Of Characters Of Finite Groups

The representation theory of finite groups is one of the powerful tools to study finite groups,such as Frobenius's theorem and p^aq^b-theorem.It is a classic and important subject to characterize finite groups by investigating the distribution of zeros in the character tables.Many scholars have studied the subject,and given many important results.In this paper,we consider the structures of finite groups by the distribution of zeros in the character tables.It consists of following three chapters:In chapter 1,we introduce some symbols and basic concepts that we usually use in the paper.Moreover,we introduce some backgrounds and results of our research.In chapter 2,we study the finite groups in which every irreducible character vanishes on at most three conjugacy classes in the character table,and get the following two theorems:Theorem 2.2.1.Let G be a finite non-abelian solvable group.If every irreducible character of G vanishes on at most three conjugacy classes,then G is just one of the following groups:(1) G is a Frobenius group with kernel G' and complement of order 2.(2) G is a Frobenius group with abelian kernel G' and complement of order 3.(3) G≌D₈ or Q₈.(4) G is a Frobenius group with kernel G' and cyclic complement of order 4.(5) G=G'P,where G' is a normal and abelian 2-complement of G,P∈Syl₂(G),|P|=4;Z(G)|=2 and G/Z(G) is a Frobenius group with kernel(G/Z(G))'≌G' and complement P/Z(G) of order 2.(6) G≌S₄. (7) G=(G')×,where is a cyclic group of order 3,t is an involution and G' is a Frobenius group with kernel G' and complement of order 2.Theorem 2.3.1.Let G be a finite non-solvable group.If every irreducible character of G vanishes on at most three conjugacy classes,then G is isomorphic to A₅, L₂(7) or A₆.In chapter 3,we mainly study two dual questions on zeros of characters of finite groups.Y.Berkovich and L.Kazarin in[2]posed the following question:classify the groups G with the following property:(*):v(x) is a conjugacy class for all but one of the non-linear irreducible characters x of G.Dually,we pose the following question:classify the groups G with the following property:(**):all but one of the columns of character table of G have at most one zero entry.The main result of this section is as follows.Theorem 3.3.1.Let G be a finite non-abelian group.Then properties(*) and (**) are equivalent,and G has these properties if and only if one of the following holds:(1) G is a 2-transitive Frobenius group with kernel G' or an extra-special 2-group.(2) G is a Frobenius group with kernel G' of order greater than 3 and complement of order 2.(3) G≌SL(2,3).(4) G≌S₄.(5) G≌A₅.At the end of the last two chapters,we leave a series of questions,which are unsolved.