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.QuestionⅠ:Classify the finite groups G such that T(X)is a conjugacy class for all but one nonlinear irreducible characters X of G; QuestionⅡ: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(?)groups.In addition,we give a complete answer to Question 1 in[4]and weaken the hypothesis of the theorem in[6].