טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentLudar Nir
SubjectGroup Characters with Many Values
DepartmentDepartment of Mathematics
Supervisor Mr. David Chillag (Deceased)


Abstract

Given a finite group G, and the set of its irreducible characters
Irr(G) = {X1, X2,…, Xn}, the properties and values of Xi are driven from  the structure of G. On the other hand, it is known that the presence of some properties and values of the characters of G affects the structure of G (even though it does not determine it uniquely).

One basic property of a character is being a class function: a character of G accepts constant values on conjugacy classes of G. In this work we study groups which have a special irreducible character X, which accepts different absolute values on different conjugacy classes. i.e. if ClG(g) is different then ClG(h) then |X(g)| is not equal to |X(h)|.

            An immediate consequence from the presence of such character in Irr(G) is the fact that G is real, meaning that every character of G accepts only real values on the elements of G. Another consequence is that G has an element x such that the order of its centralizer, CG(x), is less then 6.

            According to those two facts, most of this work is devoted to the study of real groups which has an element that has a small centralizer. We separate the cases of solvable and non-solvable groups and map the structure of all the groups that has such character.