Let G be a finite group, and Irr(G) the set of irreducible complex characters of G. We say that an element g is an element of G is a vanishing element of G if there exists chi in Irr(G) such that chi(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 Gamma(G). Focusing on the class of solvable groups, we prove that Gamma(G) has at most two connected components, and we characterize the case when it is disconnected. Moreover, we show that the diameter of Gamma(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 Gamma(G).

Dolfi, S., Pacifici, E., Sanus, L., Spiga, P. (2010). On the vanishing prime graph of solvable groups. JOURNAL OF GROUP THEORY, 13(2), 189-206 [10.1515/JGT.2009.046].

On the vanishing prime graph of solvable groups

SPIGA, PABLO
2010

Abstract

Let G be a finite group, and Irr(G) the set of irreducible complex characters of G. We say that an element g is an element of G is a vanishing element of G if there exists chi in Irr(G) such that chi(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 Gamma(G). Focusing on the class of solvable groups, we prove that Gamma(G) has at most two connected components, and we characterize the case when it is disconnected. Moreover, we show that the diameter of Gamma(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 Gamma(G).
Articolo in rivista - Articolo scientifico
Algebra and Number Theory
English
2010
13
2
189
206
reserved
Dolfi, S., Pacifici, E., Sanus, L., Spiga, P. (2010). On the vanishing prime graph of solvable groups. JOURNAL OF GROUP THEORY, 13(2), 189-206 [10.1515/JGT.2009.046].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/133178
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