Groups whose vanishing class sizes are not divisible by a given prime.
The authors establish that a finite group \(G\) whose vanishing conjugacy classes are not divisible by the prime \(p\) has a normal \(p\)-complement and Abelian Sylow \(p\)-subgroups. An element \(g\in G\) is a vanishing element of \(G\) if there is an irreducible complex character \(\chi\) with \(\chi(g)=0\). In this case, we define the vanishing conjugacy class of \(g\) in \(G\) to be the conjugacy class containing \(g\). The authors establish their main result in Theorem A. Note that this is an extension of Corollary A in a paper by \textit{S. Dolfi, E. Pacifici, L. Sanus} and \textit{P. Spiga} [J. Algebra 321, No. 1, 345-352 (2009; Zbl 1162.20005)] that showed if \(p\) is a prime number and the order of every vanishing element of \(G\) is not divisible by \(p\), then \(G\) has a normal Sylow \(p\)-subgroup. The proof of Theorem A relies upon the classification of the finite simple groups. The authors conclude the paper with examples. One such example shows that, for a finite group \(G\) whose vanishing conjugacy classes are not divisible by the prime \(p\), \(G/Z(G)\) may not necessarily be a \(p'\)-group or a Frobenius group.
scientific article; zbMATH DE number 5708652
scientific article