Facts on $ \mathbb{C} $-characters

Question

My assumption: $ G $ is a finite group & $ \chi $ is a faithful $ \mathbb{C} $-character of $ G $ with degree $ n $ and $ r $ is the number of distinct values assumed by $ \chi $. Now is it true that each irreducible $ \mathbb{C} $-character occurs as a direct summand of at least one power $ \chi^s $, when $ s= 0, 1, ..., r-1 $?

Answer 1

This is true and is a famous theorem of Burnside-Brauer. It can be found as Theorem (4.3) (http://bit.ly/1FlBmhL) in the book of I.M. Isaacs - Character Theory of Finite Groups.