Some results on conjugacy class sizes of primary elements of a finite group (Q6633629)

Let \(G\) be a finite group. If \(x \in G\) then \(o(x)\) denotes the order of \(x\) and \(x^{G}\) denotes the conjugacy class of \(x\) in \(G\). An element \(x \in G\) is primary if \(o(x)\) is a power of a prime. An element \(x\) is biprimary if \(o(x)\) is divisible by exactly two distinct primes. Let \(\operatorname{ppcs}(G)=\{|x^{G}| : x \in G \text{ and } x \text{ is biprimary} \}\). If for every \(x \not \in Z(G)\) the \(p\)-part of \(|x^{G}|\) is \(p^{\alpha}\) (\(p\) a prime and \(a \geq 1\)) the group \(G\) is called a \(PK(p^{a})\)-group. \textit{C. Casolo} et al. [Isr. J. Math. 192, Part A, 197--219 (2012; Zbl 1270.20032)] have determined the structure of \(PK(p^{\alpha})\)-groups.\N\NIn the paper under review, the authors generalize the above result by proving Theorem 2.5 Let \(G\) be a \(PK(p^{a})\)-group and let \(P\) be a Sylow \(p\)-subgroup of \(G\). Then, \(G\) is solvable and has a normal \(p\)-complement. Furthermore, \(\{|x^{P}| : x \in P \} = \{1, p^{e} \}\), \(P\) has nilpotency class at most 3 and \(P/Z(P)\) has exponent \(p\).\N\NThey also prove Theorem 2.13: Suppose that \(G\) has no non-trivial abelian direct factors and that \(\operatorname{ppcs}(G)=\{1, m, mn \}\) with \((m,n)=1\). Then, the following statements are true: \N\begin{itemize}\N\item[(1)] \(G=PH\), where \(P\) is a Sylow \(p\)-subgroup of \(G\) for some prime \(p\) and \(H\) is a normal \(p\)-complement of \(G\). \N\item[(2)] \(H\) and \(O_{p}(G)\) are abelian, \(P\) is nonabelian and \(G/O_{p}(G)\) is a Frobenius group. \N\item[(3)] \(|P/O_{p}(G)|=p\), \(|P'|= p\), \(Z(P)=Z(G)_{p}\) and \(P/Z(P)\) is an elementary abelian \(p\)-group of order \(p^{2}\). \N\item[(4)] \(m=p\) and \(n=|H|\).\N\end{itemize}