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On zeros of characters of finite groups

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Abstract

For a finite group G and a non-linear irreducible complex character χ of G write υ(χ) = {g ∈ G | χ(g) = 0}. In this paper, we study the finite non-solvable groups G such that υ(χ) consists of at most two conjugacy classes for all but one of the non-linear irreducible characters χ 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 φ-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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Authors and Affiliations

  1. School of Science, Sichuan University of Science and Engineering, Zigong, 643000, China

    Jinshan Zhang & Dandan Liu

  2. School of Mathematics, Suzhou University, Suzhou, 215006, China

    Zhencai Shen

Authors
  1. Jinshan Zhang
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  2. Zhencai Shen
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  3. Dandan Liu
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Correspondence to Jinshan Zhang.

Additional information

Project supported by the NNSF of China (Grant No. 10871032) and the NSF of Sichuan University of Science and Engineering (Grant No. 2009XJKRL011).

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Zhang, J., Shen, Z. & Liu, D. On zeros of characters of finite groups. Czech Math J 60, 801–816 (2010). https://doi.org/10.1007/s10587-010-0050-2

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  • DOI: https://doi.org/10.1007/s10587-010-0050-2

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