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On the Deskins completions, theta completions and theta pairs for maximal subgroups. II. (English) Zbl 0911.20018

Author’s abstract: This paper is a continuation of the paper reviewed above [see Zbl 0911.20017]. There we introduced the concept of \(\theta\)-completions associated with a maximal subgroup of a finite group. The concept offers a convenience for us to study the completions introduced by Deskins and gives us a way to reveal the relationship between the concepts of completions and \(\theta\)-pairs, the latter concept introduced by Mukherjee and Bhattacharya. The present paper is devoted to discussing the \(\pi\)-solvability, \(\pi\)-supersolvability and \(\pi\)-nilpotency of a finite group by using the \(\theta\)-completions. Moreover, a new proof on the Deskins conjecture concerning the supersolvability is included.

MSC:

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

Citations:

Zbl 0911.20017
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References:

[1] DOI: 10.1016/0022-4049(90)90150-G · Zbl 0699.20016
[2] DOI: 10.1080/00927879608825807 · Zbl 0892.20021
[3] DOI: 10.1016/0022-4049(86)90074-5 · Zbl 0597.20014
[4] Deskins, W.E. On maximal subgroups. Proc. Sympos. Pure Math. Vol. 1, pp.100–104. Amer. Math. Soc. · Zbl 0096.24801
[5] DOI: 10.1007/BF01188517 · Zbl 0665.20008
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[9] DOI: 10.1080/00927879508825331 · Zbl 0830.20045
[10] DOI: 10.1080/00927879808826142 · Zbl 0895.16019
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