On the vanishing prime graph of solvable groups.

*(English)*Zbl 1196.20029Let \(G\) be a finite group and \(\text{Irr}(G)\) the set of its irreducible complex characters. \(g\in G\) is called ‘vanishing’ if \(\chi(g)=0\) for some \(\chi\in\text{Irr}(G)\). In the paper under review a new graph \(\Gamma(G)\) is introduced as follows: The vertices of \(\Gamma(G)\) are the prime numbers which divide the order of at least one vanishing element of \(G\), and two such primes are connected by an edge if and only if their product divides the order of some vanishing element. For solvable \(G\) the paper then establishes some basic properties of \(\Gamma(G)\). For example, it has at most two connected components, and the groups with exactly two connected components can be characterized. Moreover, the diameter of \(\Gamma(G)\) is at most 4, and this bound is best possible. Also, there is no universal upper bound on the size of an independent set of \(\Gamma(G)\).

Reviewer: Thomas Michael Keller (San Marcos)

##### MSC:

20D60 | Arithmetic and combinatorial problems involving abstract finite groups |

20C15 | Ordinary representations and characters |

20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

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\textit{S. Dolfi} et al., J. Group Theory 13, No. 2, 189--206 (2010; Zbl 1196.20029)

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