On existence of normal \(p\)-complement of finite groups with restrictions on the conjugacy class sizes (Q6580575)

Let \(G\) be a finite group and let \(N(G)\) be the set of the sizes of non-central conjugacy classes of \(G\). If \(p \in \pi(G)\), let \(|G||_{p}\) be the number \(p^{n}\) such that \(N(G)\) contains a multiple of \(p^{n}\) and avoids multiples of \(p^{n}\) and, for \(\pi \subseteq \pi(G)\), let \(|G||_{\pi}= \prod_{p\in pi} |G||_{p}\). Let \(\Theta_{\pi}=\{ \tau \subseteq \pi \mid \tau \not =\emptyset, |\tau| \geq |\pi|-1 \}\). A group \(G\) is a \(\pi^{\ast}\)-group (or \(G \in \pi^{\ast}\)) if for every a \(a \in N(G)\) there exists \(\tau_{a} \in \Theta_{\pi}\) such that \(a_{\pi} = |G||_{\tau_{a}}\).\N\N\textit{K. Ishikawa} [Isr. J. Math. 129, 119--123 (2002; Zbl 1003.20022)] proved that a group \(G\) with \(N(G)=\{ p^{n}\}\) is nilpotent class at most \(3\). \textit{C. Casolo} et al. [Isr. J. Math. 192, Part A, 197--219 (2012; Zbl 1270.20032)] described the set of \(\{p\}^{\ast}\)-groups. In particular, they proved that any group of \(\{p\}^{\ast}\) is solvable (it is remarkable that the proof does not use CFSG) and contains a normal \(p\)-complement.\N\NIn the paper under review, the author proves that if there exists an integer \(\alpha > 0\) such that \(|G: C_{G}(g)|_{p} \in \{ 1, p^{\alpha} \}\) for every \(g \in G\) and a \(p\)-element \(x \in G\) such that \(|G: C_{G}(x)|_{p} >1\), then \(G\) has a normal \(p\)-complement.