# zbMATH — the first resource for mathematics

On zeros of characters of finite groups. (English) Zbl 0917.20007
By a well-known result of Burnside a nonlinear irreducible character over $$\mathbb{C}$$ has at least one zero on $$G$$. Here the author deals with the question how many zeros an irreducible character can have. He relates this number to the size of some centralizer. For instance, if $$G'\neq G\neq G''$$ then there exists an $$x\in G$$ such that $$| C_G(x)|\leq 2m$$ where $$m$$ denotes the maximal number of zeros in a row of the character table.
Using the classification of finite simple groups the author shows that a non-abelian group $$G$$ possessing a $$\chi\in\text{Irr}(G)$$ with at most one zero is a Frobenius group with a complement of order 2 and an abelian odd-order kernel.
Dually to the number of zeros of an irreducible character also the number of characters which vanish on a fixed conjugacy class is considered. This number is related to some character degree. For instance, if $$1\neq Z(G)\neq Z_2(G)$$ then there exists a $$\chi\in\text{Irr}(G)$$ such that $$| G|/\chi(1)^2\leq 2m$$ where $$m$$ is the maximal number of zeros in a column of the character table.

##### MSC:
 20C15 Ordinary representations and characters 20D60 Arithmetic and combinatorial problems involving abstract finite groups
Full Text: