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On the vanishing prime graph of solvable groups. (English) Zbl 1196.20029
Let \(G\) 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 \(\Gamma(G)\).

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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