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Existence of Hall normal subgroups in finite groups

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Abstract

We obtain a necessary and sufficient criterion for the existence of an invariant complement to a nilpotent subgroup contained as a direct factor in one of the maximal subgroups of a given group; we also find a condition for the p-closure of a group, all proper subgroups of which are p-closed, expressed in terms of the degree of one of its nonlinear irreducible characters.

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Literature cited

  1. 1.

    S. A. Chunikhin, Subgroups of Finite Groups, Wolters-Noordhoff, Groningen, The Netherlands (1969).

  2. 2.

    W. Feit, Characters of Finite Groups, W. A. Benjamin, New York (1967).

  3. 3.

    A. V. Romanovskii, “Groups with Hall normal divisors,” in: Finite Groups [in Russian], Nauka i Tekhnika, Minsk (1966), pp. 98–115.

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Additional information

Translated from Matematicheskie Zametki, Vol. 16, No. 3, pp. 381–385, September, 1974.

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Romanovskii, A.V. Existence of Hall normal subgroups in finite groups. Mathematical Notes of the Academy of Sciences of the USSR 16, 817–819 (1974). https://doi.org/10.1007/BF01148126

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Keywords

  • Normal Subgroup
  • Finite Group
  • Maximal Subgroup
  • Irreducible Character
  • Proper Subgroup