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On the Deskins completions, theta pairs and theta completions for maximal subgroups. (English) Zbl 0997.20026

The paper under review belongs to a series of articles devoted to the study of the influence of the maximal subgroups on the structure of the finite groups.
The tools used in this paper are the concepts of completion [W. E. Deskins, Arch. Math. 54, No. 3, 236-240 (1990; Zbl 0665.20008)] and theta pair [N. P. Mukherjee and P. Batthacharya, Proc. Am. Math. Soc. 109, No. 3, 589-596 (1990; Zbl 0699.20019)]. The author, studying the relation between them, introduced the concept of \(\theta\)-completion which is quite useful to study the embedding of maximal subgroups and clarifies the relationship between completions and \(\theta\)-pairs [Y.-Q. Zhao, Commun. Algebra 26, No. 10, 3141-3153 (1998; Zbl 0911.20017)]. This paper presents two characterisations of solubility in terms of maximal completions, maximal \(\theta\)-pairs and maximal \(\theta\)-completions for some families of maximal subgroups.

MSC:

20D25 Special subgroups (Frattini, Fitting, etc.)
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20E28 Maximal subgroups
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