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On \(c\)-normal maximal and minimal subgroups of Sylow subgroups of finite groups. II. (English) Zbl 1050.20011
[For part I cf. the first author, Commun. Algebra 29, No. 5, 2193-2200 (2001; Zbl 0990.20012).]
A subgroup \(H\) of a finite group \(G\) is said to be \(c\)-normal in \(G\) if there exists a subgroup \(N\) of \(G\) such that \(G=HN\) and \(H\cap N\) is contained in \(\text{Core}_G(H)\), the largest normal subgroup of \(G\) contained in \(H\). Given a group \(G\), the authors study the influence of the \(c\)-normality of the maximal and the minimal subgroups of the Sylow subgroups of the generalised Fitting subgroup of some normal subgroup of \(G\) on the structure of \(G\). For a saturated formation \(\mathfrak F\) containing the class of all supersoluble groups, they prove that if \(H\) is a normal subgroup of a group \(G\) such that \(G/H\in{\mathfrak F}\) and all maximal subgroups of all Sylow subgroups of the generalised Fitting subgroup \(F^*(H)\) of \(H\) are \(c\)-normal in \(G\), then \(G\in{\mathfrak F}\), and that if all subgroups of order a prime number or \(4\) of \(F^*(H)\) are \(c\)-normal in \(G\), then \(G\in{\mathfrak F}\).
Reviewer’s remark: There are some misprints in the paper, e.g., in the statement of Theorem 3.2, where the \(c\)-normality is imposed on all minimal subgroups and all cyclic subgroups of \(F^*(H)\), but it is actually needed only for all minimal subgroups and all cyclic subgroups of order \(4\) of \(F^*(H)\).

MSC:
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D40 Products of subgroups of abstract finite groups
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