\(p\)-divisibility of conjugacy class sizes and normal \(p\)-complements.
Let \(G\) be a finite group and \(p\) be a prime. By a theorem of \textit{J.~G.~Thompson} [J. Algebra 14, 129-134 (1970; Zbl 0205.32606)], if the degree of every nonlinear irreducible character of \(G\) is divisible by \(p\), then \(G\) has a normal \(p\)-complement. The analogue statement for conjugacy classes of this theorem, that is, if every noncentral conjugacy class has size divisible by \(p\), then the group has normal \(p\)-complement, is not true in general. Moreover, even if \(G\) is a solvable group with this property, it can have arbitrary large \(p\)-length (see [\textit{B. Huppert}, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 9, No. 3, 145-148 (1998; Zbl 0930.20016)]). In contrast to this negative result, it was proved by \textit{C. Casolo, S. Dolfi} and \textit{E. Jabara} [Isr. J. Math. 192, Part A, 197-219 (2012; Zbl 1270.20032)] that if every noncentral conjugacy class size of \(G\) has the same \(p\)-part, then \(G\) is solvable and it has a normal \(p\)-complement. The paper under review presents a twofold extension of this result: The authors prove that if \(N\) is any normal subgroup of \(G\) and \(|x^G|\) has the same \(p\)-part for every \(x\in N\setminus Z(N)\) of prime power order, then \(N\) is solvable and it has a normal \(p\)-complement. Unlike to the proof of Casolo et al., the proof of this result depends on the classification theorem of the finite simple groups.
scientific article; zbMATH DE number 6387039