Quasinormal Fitting classes of finite groups

  • Anna V. Martsinkevich P. M. Masherov Vitebsk State University, 33 Maskoŭski Avenue, Vitebsk 210038, Belarus
DOI: https://doi.org/10.33581/2520-6508-2019-2-18-26

Abstract

Let P be the set of all primes, Zn a cyclic group of order n and X wr Zn the regular wreath product of the group X with Zn. A Fitting class F is said to be X-quasinormal (or quasinormal in a class of groups X ) if F ⊆ X, p is a prime, groups G ∈ F and G wr Zp ∈ X, then there exists a natural number m such that G m wr Zp ∈ F. If  X is the class of all soluble groups, then F is normal Fitting class. In this paper we generalize the well-known theorem of Blessenohl and Gaschütz in the theory of normal Fitting classes. It is proved, that the intersection of any set of nontrivial X-quasinormal Fitting classes is a nontrivial X-quasinormal Fitting class. In particular, there exists the smallest nontrivial X-quasinormal Fitting class. We confirm a generalized version of the Lockett conjecture (in particular, the Lockett conjecture) about the structure of a Fitting class for the case of X-quasinormal classes, where X is a local Fitting class of partially soluble groups.

Author Biography

Anna V. Martsinkevich, P. M. Masherov Vitebsk State University, 33 Maskoŭski Avenue, Vitebsk 210038, Belarus

postgraduate student at the department of algebra and methods of teaching mathematics, faculty of mathematics and information technology

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Published
2019-07-15
Keywords: Fitting class, quasinormal Fitting class, the Lockett conjecture, local Fitting class
Supporting Agencies Research is supported by the Belarusian Republican Foundation for Fundamental Research (grant No. Ф17M-064). The author would like to express sincere gratitude to professor N. T. Vorob’ev for the formulation of the problem and discussion of the results of work.
How to Cite
Martsinkevich, A. V. (2019). Quasinormal Fitting classes of finite groups. Journal of the Belarusian State University. Mathematics and Informatics, 2, 18-26. https://doi.org/10.33581/2520-6508-2019-2-18-26
Issue
No 2 (2019): Journal of the Belarusian State University. Mathematics and Informatics
Section
Mathematical Logic, Algebra and Number Theory

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