A note on finite groups in which every non-nilpotent maximal subgroup has prime index (Q6601862)

Let \(G\) be a finite group. A well-known result by \textit{B. Huppert} [Math. Z. 60, 409--434 (1954; Zbl 0057.25303)] asserts that if every maximal subgroup of \(G\) has prime index, then \(G\) is supersolvable. As a partial generalization of Huppert's theorem, it was shown by \textit{J. Lu} et al. [Monatsh. Math. 171, No. 3--4, 425--431 (2013; Zbl 1277.20017)] that if every non-nilpotent maximal subgroup of \(G\) has prime index, then \(G\) is solvable (but the proof requires some results from the classification of finite simple groups).\N\NIn the paper under review, without using the solvability of a group \(G\) in which every non-nilpotent maximal subgroup has prime index, the authors obtain the following structural property for \(G\) (see Theorem 1.1): Let \(G\) be a finite group and let \(p\) be the largest prime divisor of \(|G|\). If every non-nilpotent maximal subgroup of \(G\) has prime index, then either the Sylow \(p\)-subgroup of \(G\) is normal or \(G\) has a normal \(p\)-complement.