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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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