On the \(\mathcal F\)-abnormal maximal subgroups of finite groups. (Q856354)

For any saturated formation \(\mathcal F\) of finite groups containing all supersolvable groups, the groups in \(\mathcal F\) are characterized in the paper by the \(\mathcal F\)-abnormal maximal subgroups. Let \(\mathcal F\) be a saturated formation of finite groups and let \(\mathcal U\) be the class of all finite supersolvable groups. Let \(G\) be a finite group. A maximal subgroup \(M\) of \(G\) is called \(\mathcal F\)-normal in \(G\) if \(G/M_G\in{\mathcal F}\), where \(M_G\) denotes the core of \(M\) in \(G\); otherwise \(M\) is said to be \(\mathcal F\)-abnormal in \(G\). Let \({\mathcal M}_c (G)\) denote the set of \(\mathcal F\)-abnormal maximal subgroups \(M\) of \(G\) such that \(|G:M|\) is composite. Let \(\mathcal F\) be a saturated formation containing \(\mathcal U\). In the paper, the authors give the definition of \(s\)-\(\theta\)-completion for the first time and prove the following main results: (i) \(G\in{\mathcal F}\) if and only if every \(\mathcal F\)-abnormal maximal subgroup of \(G\) has index a prime; (ii) \(G\in{\mathcal F}\) if and only if, for every \(M\in{\mathcal M}_c(G)\), \(M\) has an \(s\)-\(\theta\)-completion \(C\) such that \(G=CM\) and \(C/M_G\) has square free order; (iii) Let \(G\) be a finite group which is \(S_4\)-free. Then \(G\in{\mathcal F}\) if and only if every \(M\in{\mathcal M}_c(G)\) has an \(s\)-\(\theta\)-completion \(C\) such that \(C/M_G\) is cyclic with \(|C/M_G|\geq|G:M|\).