Abstract
Let G be a finite group. An element \({g\in G}\) is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy class of G. In this paper we show that, if the size of every vanishing conjugacy class of G is not divisible by a given prime number p, then G has a normal p-complement and abelian Sylow p-subgroups.
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Dolfi, S., Pacifici, E. & Sanus, L. Groups whose vanishing class sizes are not divisible by a given prime. Arch. Math. 94, 311–317 (2010). https://doi.org/10.1007/s00013-010-0107-3
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DOI: https://doi.org/10.1007/s00013-010-0107-3