有限群的特征标的零点和可解φ-群

有两个对偶的问题如下：问题Ⅰ：将满足下述条件的有限群G分类：G的特征标表中，除一行外其余各行最多有一个零．问题Ⅱ：将满足下述条件的有限群G分类：G的特征标表中，除一列外其余各列最多有一个零．在这篇文章中，我们对于有限可解群解答上述两个问题，并确定和这两个问题密切相关的一类有限可解群的结构（这类可解群在本文中称之为可解φ-群）．附带我们还完全回答了4]中的问题1，并说明6，定理]的条件可以极大地减弱．

On Zeros of Characters of Finite Groups and Solvable φ-groups

Let G be a finite group, and let X be a nonlinear irreducible character of G. We set T(X) = {g ∈ G|x(g) = 0}. There are two dual questions as follows. Question I: Classify the finite groups G such that T(X) is a conjugacy class for all but one nonlinear irreducible characters X of G;Question If: Classify the finite groups G such that there is at most one zero entry in all but one columns of the character table of G. In this paper, we answer these two questions for finite solvable groups,and determine the structures of a class of finite solvable groups which are closely related to the above-mentioned two questions and are called solvable φ-groups. In addition, we give a complete answer to Question 1 in 4] and weaken the hypothesis of the theorem in 6].