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Finite groups with seminormal Schmidt subgroups

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Algebra and Logic Aims and scope

Abstract

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB1 is a proper subgroup of G, for every proper subgroup B1 of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary.

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Authors and Affiliations

  1. Gomel Engineering Institute under the Belarussian Ministry of Extraordinary Situations, Belarus

    V. N. Knyagina

  2. Gomel State University, Gomel, Belarus

    V. S. Monakhov

Authors
  1. V. N. Knyagina
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  2. V. S. Monakhov
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Additional information

Supported by BelFBR grant Nos. F05-341 and F06MS-017.

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Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.

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Knyagina, V.N., Monakhov, V.S. Finite groups with seminormal Schmidt subgroups. Algebr Logic 46, 244–249 (2007). https://doi.org/10.1007/s10469-007-0023-1

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  • DOI: https://doi.org/10.1007/s10469-007-0023-1

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