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Groups whose vanishing class sizes are not divisible by a given prime

Authors:	Silvio Dolfi Emanuele Pacifici Lucia Sanus

Affiliation:	1. Dipartimento di Matematica U. Dini, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy 2. Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, Via Saldini 50, 20133, Milan, Italy 3. Departament d’àlgebra, Facultat de Matemàtiques, Universitat de València, 46100, Burjassot, Valencia, Spain

Abstract:	Let G be a finite group. An element ${gin G}Let G be a finite group. An element g ? G{gin 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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