On zeros of characters of finite groups.

*(English)*Zbl 0917.20007By 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.

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.

Reviewer: W.Willems (Magdeburg)

##### MSC:

20C15 | Ordinary representations and characters |

20D60 | Arithmetic and combinatorial problems involving abstract finite groups |