Groups whose vanishing class sizes are prime powers
Let \(G\) be a finite group. An element \(g \in G\) (and the conjugacy class \(g^{G}\)) is called vanishing if there is an irreducible character \(\chi \in \mathrm{Irr}(G)\) such that \(\chi(g)=0\). It is a well-known fact that a prime \(p \in \pi(G)\) does not divide any conjugacy class size of \(G\) if and only if \(G\) has a central Sylow \(p\)-subgroup. In view of this result, \textit{S. Dolfi} et al. [Arch. Math. 94, No. 4, 311--317 (2010; Zbl 1202.20010)] have proven that if \(G\) has no vanishing classes of size divisible by a prime \(p\), then \(G\) has a normal \(p\)-complement and abelian Sylow \(p\)-subgroups. In the paper under review, the authors describe finite groups whose vanishing class sizes are all prime powers (see Theorem 1.1). As a corollary they prove that if \(G\) is finite group whose vanishing class sizes are prime powers and \(x \in G\) is a non-vanishing element, then \(x \in F(G)\).
scientific article; zbMATH DE number 7459652
scientific article