M.Sc Student Maayan Hila On Zeroes of Characters of Finite Groups Department of Mathematics Mr. David Chillag (Deceased) Abstract

This paper deals with zeroes of irreducible characters of finite groups, and with various conclusions drawn from these to the structure of the group. All groups in this paper are finite.

A character of a group G is a class function, i.e., it is constant on all the elements in a joint conjugacy class of G. The values of a character are complex numbers. A finite group G has an infinite number of characters, but a finite number of irreducible characters. These irreducible characters form a basis of the space of class functions on the group. The number of irreducible characters of G is the same as the number of conjugacy classes in G. The irreducible characters of the group G say a lot about the structure of G as a group.

A rational group is a group whose characters attain only rational values.

The degree of a character is its value on the unit element. This is always a natural number, and when it is 1, the character is called linear.

In this paper we shall research rational groups G satisfying the following condition: all non-linear irreducible characters of G satisfy that the degree of the character multiplied by the number of zeroes of the character produces the order of G.

We will show that groups G satisfying the above condition are necessarily solvable. To this end we will first prove that [G,G] is necessarily a non-rational group.

We will also show that there are no non-abelian 2-groups satisfying the above condition, and, in fact, we will prove that the only prime numbers appearing in the factorization of the order of such a group G are 2 and 3. Moreover, both 2 and 3 must appear.

The main result of this paper is a detailed description of all finite rational groups satisfying this condition.