The Number Of Zeros In The Character Table And The Structure Of Finite Group

Let G be a finite group. If there are very little zero points in the character table of G, then the structure of G can be restricted. There were many outcomes about determining the structure of the group by the distribution of zero points in its character table. In this paper, we consider the group satisfying one of the following conditions:(1) at most one zero in every column of the character table of finite group G, we say such a group is V(l)-group;(2) at most two zeros in every column of the character table of finite group G, we say such a group is V(2)-group;(3) at most p zeros in every column of the character table of finite group G (where p is the minimal prime divisor of JG|), we say such a group is V(p)-group;We get the following theorems:Theorem 3.1 A finite group G is a V(1)-group if and only if one of the following occurs(1) G is abelian;(2) G is an extraspecial 2-group;(3) G = H_∝ N is a Frobenius group with kernel N and complement H where H is abelian and N is an elementary abelian p-group and |H| — |N| — 1.Theorem 3.2 Let G be a solvable finite group. Then G is a V(2)-group but not V(l)-group if and only if one of the following occurs (1) G (?) S₄;...