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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W.

Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus

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Volume 13, Issue 2


On the vanishing prime graph of solvable groups

Silvio Dolfi
  • Dipartimento di Matematica U. Dini, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy. E-mail:
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/ Emanuele Pacifici
  • Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy. E-mail:
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/ Lucia Sanus
  • Departament d'Àlgebra, Facultat de Matemátiques, Universitat de València, 46100 Burjassot, València, Spain. E-mail:
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/ Pablo Spiga
  • Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, via Trieste 63, 35121 Padova, Italy. E-mail:
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Published Online: 2009-08-31 | DOI: https://doi.org/10.1515/jgt.2009.046


Let G be a finite group, and Irr(G) the set of irreducible complex characters of G. We say that an element gG is a vanishing element of G if there exists χ in Irr(G) such that χ(g) = 0. In this paper, we consider the set of orders of the vanishing elements of a group G, and we define the prime graph on it, which we denote by Γ(G). Focusing on the class of solvable groups, we prove that Γ(G) has at most two connected components, and we characterize the case when it is disconnected. Moreover, we show that the diameter of Γ(G) is at most 4. Examples are given to round out our understanding of this matter. Among other things, we prove that the bound on the diameter is best possible, and we construct an infinite family of examples showing that there is no universal upper bound on the size of an independent set of Γ(G).

About the article

Received: 2009-02-06

Revised: 2009-06-17

Published Online: 2009-08-31

Published in Print: 2010-03-01

Citation Information: Journal of Group Theory, Volume 13, Issue 2, Pages 189–206, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/jgt.2009.046.

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