Geodesic
On zeros of characters of finite groups
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 801-816 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 20C15
Keywords: finite groups; characters; zeros 
@article{CMJ_2010_60_3_a13,
     author = {Zhang, Jinshan and Shen, Zhencai and Liu, Dandan},
     title = {On zeros of characters of finite groups},
     journal = {Czechoslovak Mathematical Journal},
     pages = {801--816},
     year = {2010},
     volume = {60},
     number = {3},
     mrnumber = {2672416},
     zbl = {1208.20005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a13/}
}
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Zhang, Jinshan; Shen, Zhencai; Liu, Dandan. On zeros of characters of finite groups. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 801-816. http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a13/