On the vanishing prime graph of solvable groups (original) (raw)
2010, Journal of Group Theory
Let G be a finite group, and IrrðGÞ the set of irreducible complex characters of G. We say that an element g A G is a vanishing element of G if there exists w in IrrðGÞ such that wðgÞ ¼ 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 GðGÞ. Focusing on the class of solvable groups, we prove that GðGÞ has at most two connected components, and we characterize the case when it is disconnected. Moreover, we show that the diameter of GðGÞ 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 GðGÞ.
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