A finite group with a maximal Miller-Moreno subgroup (Q6576791)

A Miller-Moreno group is a finite non-abelian group all of whose proper subgroups are abelian (see [\textit{G. A. Miller} and \textit{H. C. Moreno}, A. M. S. Trans. 4, 398--404 (1903; JFM 34.0173.01)]). A well-known result of \textit{J. G. Thompson} [Proc. Natl. Acad. Sci. USA 45, 578--581 (1959; Zbl 0086.25101)] asserts that if \(G\) is a finite group with a nilpotent maximal subgroup of odd order, then \(G\) is solvable.\N\NIn the paper under review, the authors study a finite group in which some maximal subgroup is a Miller-Moreno group. Let \(G\) be a solvable group and let \(d\ell(G)\) denote the derived length of \(G\). The main result is Theorem 1: Let \(G\) be a finite group and let \(M\) be its maximal subgroup. Suppose that \(M\) is a Miller-Moreno group. Then the following assertions hold: \N\begin{itemize}\N\item[(a)] if \(G\) is solvable, then \(|\pi(G)| \leq 3\) and \(d\ell(G) \leq 4\); moreover, if \(|\pi(G)|=3\), then \(G\) has a Sylow tower and \(d\ell(G) \leq 3\); \N\item[(b)] if \(G\) is unsolvable, then \(Z(G)=Z(M)\) and \(G/Z(G)\) is one of the simple groups \(A_{p}\), \(L_{2}(q)\), \(L_{n}^{\epsilon}(q)\), \(^{2}B_{2}(q)\), \(M_{23}\), \(\mathbb{B}\) (where the conditions on \(p\) and \(q\) are specified in Table 1 of the paper).\N\end{itemize}