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Keywords:
finite groups; characters; zeros
finite groups; characters; zeros
Summary:
For a finite group $G$ and a non-linear irreducible complex character $\chi $ of $G$ write $\upsilon (\chi )=\{g\in G\mid \chi (g)=0\}$. In this paper, we study the finite non-solvable groups $G$ such that $\upsilon (\chi )$ consists of at most two conjugacy classes for all but one of the non-linear irreducible characters $\chi $ of $G$. In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable $\varphi $-groups. As a corollary, we answer Research Problem $2$ in [Y. Berkovich and L. Kazarin: Finite groups in which the zeros of every non-linear irreducible character are conjugate modulo its kernel. Houston J. Math.\ 24 (1998), 619--630.] posed by Y. Berkovich and L. Kazarin.
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