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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Maximal subgroups of prime index in a finite solvable group


Author: Paul Venzke
Journal: Proc. Amer. Math. Soc. 68 (1978), 140-142
MSC: Primary 20D10
MathSciNet review: 0476851
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Abstract: In this note we show that a maximal subgroup of a finite solvable group has prime index if and only if it admits a cyclic supplement which permutes with one of its Sylow systems. In particular, a finite solvable group is supersolvable if and only if each maximal subgroup admits a cyclic supplement which permutes with a Sylow system of the maximal subgroup.


References [Enhancements On Off] (What's this?)

  • [1] Bertram Huppert, Normalteiler und maximale Untergruppen endlicher Gruppen, Math. Z. 60 (1954), 409–434 (German). MR 0064771
  • [2] Otto H. Kegel, On Huppert’s characterization of finite supersoluble groups, Proc. Internat. Conf. Theory of Groups (Canberra, 1965) Gordon and Breach, New York, 1967, pp. 209–215. MR 0217183
  • [3] Paul Venzke, System quasinormalizers in finite solvable groups, J. Algebra 44 (1977), no. 1, 160–168. MR 0437638

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0476851-2
Keywords: Maximal subgroup, supersolvable, quasinormal
Article copyright: © Copyright 1978 American Mathematical Society