On vanishing criteria that control finite group structure (Q290427)

A conjugacy class in a finite group \(G\) is called vanishing if there exists some irreducible character of \(G\) which evaluates to zero on the conjugacy class.NEWLINENEWLINE In this paper, there are few results proven regarding vanishing conjugacy classes:NEWLINENEWLINE A) Let \(p\) be a certain prime dividing \(|G|\) in such a way that, if \(q\) is an arbitrary prime dividing \(|G|\), it always holds that \(q\) does not divide \(p-1\). Then \(G\) is solvable if no number \(|C|\) is divisible by \(p^2\) where \(C\) is any vanishing conjugacy class of \(G\).NEWLINENEWLINE B) Suppose every vanishing conjugacy class \(C\) of \(G\) has its cardinality \(|C|\) square-free. Then \(G\) is supersolvable.NEWLINENEWLINE C) Let \(x\) be a \(p'\)-element of \(G\), \(p\) a given prime. Suppose that in this situation \(x^G\) is a vanishing conjugacy class and that \(|x^G|\) is not divisible by \(p\). Then \(G\) has a normal \(p\)-complement.NEWLINENEWLINE In order to prove those assertentions, the author deals with ordinary and modular characters, characters of defect zero, finite simple groups, formations of groups, Fitting groups etc.NEWLINENEWLINE Several related very recent results extended and expanded or otherwise generalized. As such we mention papers and results therein by \textit{A. R. Camina} [J. Lond. Math. Soc., II. Ser. 5, 127--132 (1972; Zbl 0242.20025)], \textit{J. Cassey} and \textit{Y. Wang} [Commun. Algebra, 4347--4353 (1999; Zbl 0948.20010)], \textit{S. Dolfi} et al. [Arch. Math. 94, No. 4, 311--317 (2010; Zbl 1202.20010)], and \textit{S. Dolfi} et al. [J. Algebra 323, No. 2, 540--545 (2010; Zbl 1196.20029)].NEWLINENEWLINE Of course, questions on sizes of conjugacy classes have to be brought into correspondence with similar questions on characters degrees, mentioned in the papers above and in other papers in the list of references.NEWLINENEWLINE In conclusion, a nice paper to be studied.