## On maximal subgroups of finite groups and theta pairs.(English)Zbl 0892.20021

This paper presents further investigations on $$\theta$$-pairs and completions in finite groups, concepts introduced by N. Mukherjee and P. Bhattacharya and by the reviewer. In the first section the authors exhibit the relationship between completions and $$\theta$$-pairs. Then they obtain necessary and sufficient conditions for solvability in terms of $$\theta$$-pairs (Theorems 2 and 4) and they study the intrinsic properties of a given maximal subgroup $$M$$ of finite group $$G$$ and its associated $$\theta$$-pairs which imply $$G$$ to be solvable, $$\pi$$-solvable or supersolvable (Theorems 1, 3 and 5).

### MSC:

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

 [1] DOI: 10.1016/0022-4049(90)90150-G · Zbl 0699.20016 [2] Ballester-Bolinches A., Siberian Math. Journal 64 (1990) [3] DOI: 10.1090/S0002-9939-1992-1088437-7 [4] Beidleman J.C., Illinois J.Math 16 pp 95– (1972) [5] Deskins, W.E. On maximal suboroups. Proc.Sympos.Pure Math. Providence, R.I. Vol. l, pp.100–104. Amer.Math. Soc. [6] DOI: 10.1007/BF01188517 · Zbl 0665.20008 [7] DOI: 10.1515/9783110870138 [8] Ruppert B., Endliche Gruppen I (1967) [9] DOI: 10.1090/S0002-9939-1989-0952319-9 [10] DOI: 10.1090/S0002-9939-1990-1015683-9 [11] DOI: 10.1090/S0002-9947-1971-0284495-9 [12] DOI: 10.1080/00927879408825094 · Zbl 0814.20016 [13] DOI: 10.1080/00927879508825331 · Zbl 0830.20045 [14] Zhao Y., J.Pure Appl.Algebra 23 (1995)
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