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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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On zeros of characters of finite groups
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by David Chillag

Proc. Amer. Math. Soc. 127 (1999), 977-983

DOI: https://doi.org/10.1090/S0002-9939-99-04790-5

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Abstract:

We present several results connecting the number of conjugacy classes of a finite group on which an irreducible character vanishes, and the size of some centralizer of an element. For example, we show that if $G$ is a finite group such that $G\ne G’\ne G''$, then $G$ has an element $x$, such that $|C_G(x)|\le 2m$, where $m$ is the maximal number of zeros in a row of the character table of $G$. Dual results connecting the number of irreducible characters which are zero on a fixed conjugacy class, and the degree of some irreducible character, are included too. For example, the dual of the above result is the following: Let $G$ be a finite group such that $1\ne Z(G)\ne Z_2(G)$; then $G$ has an irreducible character $\chi$ such that $\frac {|G|}{\chi ^2(1)}\le 2m$, where $m$ is the maximal number of zeros in a column of the character table of $G$.

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