M.Sc Thesis
M.Sc Student Kofman Irena On Zeros of Characters of Finite Groups Department of Mathematics Mr. David Chillag (Deceased)

Abstract

This research deals with the zeros of characters of finite groups, and what can be said on the group from the way the zeros are spread in the charter table.

A character of a finite group G is a function from the group to C, the field of complex numbers.

A finite group G has an infinite number of characters, but only a finite numbers of irreducible characters, and every non-linear irreducible character vanishes on at least one conjugacy class. We arrange the values of all irreducible characters in a table, called the character table of? G.? Restricting the character values and the way they are spread in the character table restricts the group itself. In particular, we will restrict the number of zeros of the irreducible characters and the way they are spread in the charter table. The number of zeros of? an irreducible character is the number of conjugacy classes on which it vanishes.

In the article by? D. Chillag, A. Gillio and M. Bianchi,? groups which all its irreducible non-linear characters vanishe on at most two conjugacy classes is described. Irit Orgil in her thesis has described the groups with the property that all irreducible non-linear characters, except possibly one, vanish on only one conjugacy class.

In this research we will describe the groups with the property that all irreducible non-linear characters, exept possibly two, vanish on only one conjugacy class, and there is at most one irreducible non-linear character that vanishes on four or more conjugacy classes. In addition, we will describe all solvable groups with up to ?zeros in charter table, where? k? is the number of irreducible non-linear characters; and also we formulate and prove the dual theorem to the theorem of? Irit Orgil.