Formations of finite groups in polynomial time: the F-radical and the F-length

Authors

  • Viachaslau I. Murashka Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus

Keywords:

finite group, permutation group computation, Baer-local formation, Fitting formation, F-radical, F-length, polynomial time algorithm
Supporting Agencies
This work is supported by the Belarusian Republican Foundation for Fundamental Research (F23RNF-237).

Abstract

For a Baer-local (composition) Fitting formation F of finite groups the algorithm for the computation of the F-radical of a permutation finite group which runs in polynomial time from its degree is herein suggested. It is shown how one can compute the F-radical in case when F is a primitive saturated formation of soluble finite groups. The algorithms for the computation of different lengths associated with a finite group (the generalised Fitting height, the non-p-soluble length and etc.) are presented. In the case of a permutation group these algorithms run in polynomial time from its degree.

Author Biography

  • Viachaslau I. Murashka, Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus

    PhD (physics and mathematics), docent; associate professor at the department of algebra and geometry, faculty of mathematics and technologies of programming, and leading researcher of research sector

References

  1. Bryce RA, Cossey J. Subgroup closed Fitting classes are formations. Mathematical Proceedings of the Cambridge Philosophical Society. 1982;91(2):225–258. DOI: 10.1017/S0305004100059272.
  2. Berger TR, Bryce RA, Cossey J. Quotient closed metanilpotent Fitting classes. Journal of the Australian Mathematical Society. 1985;38(2):157–163. DOI: 10.1017/S1446788700023004.
  3. Höfling B. Computing projectors, injectors, residuals and radicals of finite soluble groups. Journal of Symbolic Computation. 2001;32(5):499–511. DOI: 10.1006/jsco.2001.0477.
  4. Eick B, Wright CRB. Computing subgroups by exhibition in finite solvable groups. Journal of Symbolic Computation. 2002;33(2):129–143. DOI: 10.1006/jsco.2000.0503.
  5. Murashka VI. Formations of finite groups in polynomial time: F-residuals and F-subnormality. Journal of Symbolic Computation. 2024;122:102271. DOI: 10.1016/j.jsc.2023.102271.
  6. Shemetkov LA. Composition formations and radicals of finite groups. Ukrainian Mathematical Journal. 1988;40:318–322. DOI: 10.1007/BF01061312.
  7. Doerk K, Hawkes T. Finite soluble groups. Berlin: Walter de Gruyter; 1992. 893 p. (De Gruyter expositions in mathematics; volume 4). DOI: 10.1515/9783110870138.
  8. Guo W. Structure theory for canonical classes of finite groups. Berlin: Springer; 2015. 359 p. DOI: 10.1007/978-3-662-45747-4.
  9. Seress Á. Permutation group algorithms. Cambridge: Cambridge University Press; 2003. 264 p. DOI: 10.1017/CBO9780511546549.
  10. Neumann PM. Some algorithms for computing with finite permutation groups. In: Robertson EF, Campbell CM, editors. Proceedings of groups – St. Andrews 1985. Cambridge: Cambridge University Press; 1987. p. 59–92 (London Mathematical Society Lecture Note Series; volume 121). DOI: 10.1017/CBO9780511600647.006.
  11. Kantor WM, Luks EM. Computing in quotient groups. In: Ortiz H, editor. Proceedings of the twenty-second annual ACM symposium on theory of computing; 1990 May 13–17; Baltimore, Maryland, USA. New York: Association for Computing Machinery; 1990. p. 524–534. DOI: 10.1145/100216.100290.
  12. Babai L. On the length of subgroup chains in the symmetric group. Communications in Algebra. 1986;14(9):1729–1736. DOI: 10.1080/00927878608823393.
  13. Bryce RA, Cossey J. Fitting formations of finite soluble groups. Mathematische Zeitschrift. 1972;127(3):217–223. DOI: 10.1007/BF01114925.
  14. Shemetkov LA. Formations of finite groups. Moscow: Nauka; 1978. 272 p. Russian.
  15. Makan AR. The Fitting length of a finite soluble group and the number of conjugacy classes of its maximal metanilpotent subgroups. Canadian Mathematical Bulletin. 1973;16(2):233–237. DOI: 10.4153/CMB-1973-040-3.
  16. Khukhro EI, Shumyatsky P. Nonsoluble and non-p-soluble length of finite groups. Israel Journal of Mathematics. 2015;207(2):507–525. DOI: 10.1007/s11856-015-1180-x.
  17. Khukhro EI, Shumyatsky P. On the length of finite factorized groups. Annali di Matematica Pura ed Applicata. 2015;194(6):1775–1780. DOI: 10.1007/s10231-014-0443-1.
  18. Murashka VI, Vasil’ev AF. On the lengths of mutually permutable products of finite groups. Acta Mathematica Hungarica. 2023;170(1):412–429. DOI: 10.1007/s10474-023-01346-2.
  19. Murashka VI. Formations of finite groups in polynomial time II: the F-hypercenter and its generalizations. Trudy Instituta matematiki i mekhaniki UrO RAN. 2025;31(1):154–165. DOI: 10.21538/0134-4889-2025-31-1-154-165.
  20. Skiba AN. On some classes of sublattices of the subgroup lattice. Journal of the Belarusian State University. Mathematics and Informatics. 2019;3:35–47. DOI: 10.33581/2520-6508-2019-3-35-47.
  21. Safonov VG, Safonova IN, Skiba AN. On Baer-σ-local formations of finite groups. Communications in Algebra. 2020;48(9):4002–4012. DOI: 10.1080/00927872.2020.1753760.
  22. Safonova IN, Safonov VG. On some properties of the lattice of totally σ-local formations of finite groups. Journal of the Belarusian State University. Mathematics and Informatics. 2020;3:6–16. DOI: 10.33581/2520-6508-2020-3-6-16.

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Published

2025-06-10

Issue

No. 1 (2025): Journal of the Belarusian State University. Mathematics and Informatics

Section

Mathematical Logic, Algebra and Number Theory

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How to Cite

[1]
Murashka, V.I. 2025. Formations of finite groups in polynomial time: the F-radical and the F-length. Journal of the Belarusian State University. Mathematics and Informatics. 1 (Jun. 2025), 14–22.