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A generalisation of Kramer’s theorem and its applications. (English) Zbl 1012.20010
A theorem of Kramer states that a finite soluble group \(G\) is supersoluble if and only if, for every maximal subgroup \(M\) of \(G\), either the Fitting subgroup \(F(G)\) of \(G\) is contained in \(M\), or \(M\cap F(G)\) is a maximal subgroup of \(F(G)\). The main result of this paper (Theorem 3.1) generalises this theorem to saturated formations \(\mathfrak F\) containing the class of all supersoluble groups: If \(\mathfrak F\) is a saturated formation containing the class of all supersoluble groups, \(H\) is a soluble normal subgroup of \(G\) and \(G/H\in{\mathfrak F}\), and for every maximal subgroup \(M\) of \(G\), either \(F(H)\leq M\) or \(F(H)\cap M\) is a maximal subgroup of \(F(H)\), then \(G\in{\mathfrak F}\). The converse holds when \(\mathfrak F\) is the class of all supersoluble groups.
On the other hand, in [A. Ballester-Bolinches, Y. Wang, X. Guo, Glasg. Math. J. 42, No. 3, 383-389 (2000; Zbl 0968.20009)] the concept of \(c\)-supplementation was introduced as follows: A subgroup \(H\) of a finite group \(G\) is said to be \(c\)-supplemented in \(G\) if there exists a subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K\leq\text{Core}_G(H)\). If \(\mathfrak F\) is a saturated formation containing the class of all supersoluble groups, and \(G\) is a group with a soluble normal subgroup \(H\) such that \(G/H\in{\mathfrak F}\), and all minimal subgroups and all cyclic subgroups of order \(4\) of \(F(H)\) are \(c\)-supplemented in \(G\), then \(G\in{\mathfrak F}\) (Theorem 4.1). If \(G\) is a group with a soluble normal subgroup \(H\) such that \(G/H\in{\mathfrak F}\), and all maximal subgroups of all Sylow subgroups of \(F(H)\) are \(c\)-supplemented in \(G\), then \(G\in{\mathfrak F}\) (Theorem 4.5). As corollaries, some known sufficient conditions for supersolubility related to \(c\)-supplementation, complementation, or \(c\)-normality of distinguished subgroups of a group are recovered.

MSC:
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D40 Products of subgroups of abstract finite groups
20D25 Special subgroups (Frattini, Fitting, etc.)
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20E28 Maximal subgroups
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