рус eng
«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2024. Vol 47

On Direct Products of Dihedral Groups in Locally Finite Groups

Author(s)
Ivan A. Timofeenko1, Aleksei A. Shlepkin1

1Siberian federal university, Krasnoyarsk, Russian Federation

Abstract
When studying infinite groups, as a rule, some finiteness conditions are imposed. For example, they require that the group be periodic, a Shunkov group, a Frobenius group, or a locally finite group. The concept of saturation allows us to effectively establish the internal structure of various classes of infinite groups. To date, a large array of results on groups saturated with various classes of finite groups has been obtained. Another important direction in the study of groups with saturation conditions is the study of groups saturated by direct products of various groups. Significant progress has been made in solving the problem of B. Amberg and L. S. Kazarin on periodic groups saturated with dihedral groups in the class of locally finite groups. It is proved that a locally finite group saturated by the direct product of a finite number of finite groups of a dihedron is isomorphic to the direct product of locally cyclic groups multiplied by an involution. It is also proved that a locally finite group saturated by the direct product of a finite number of finite dihedral groups is solvable.
About the Authors

Ivan A. Timofeenko, Cand. Sci. (Phys.-Math.), Head of Laboratory, Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, ivan.timofeenko@gmail.com

Aleksei A. Shlepkin, Dr. Sci. (Phys.-Math.), Prof., Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, shlyopkin@mail.ru

For citation
Timofeenko I. A., Shlepkin A. A. On Direct Products of Dihedral Groups in Locally Finite Groups. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 47, pp. 137–146. (in Russian) https://doi.org/10.26516/1997-7670.2024.47.137
Keywords
locally finite group, direct product of groups, dihedral group, saturation by a given set of groups
UDC
512.542
MSC
20E25
DOI
https://doi.org/10.26516/1997-7670.2024.47.137
References
  1. Belousov I.N., Kondratiev A.S., Rozhkov A.V. XII school conference dedicated to the 65th anniversary of the birth of A.A. Makhneva. Tr. IMM UrO RAN, 2018, vol. 24, no. 3, pp. 281–285. https://doi.org/10.21538/0134-4889-2018-24-3-286-295 (in Russian)
  2. Kukharev A.V., Shlepkin A.A. Locally Finite Groups Saturated with Direct Product of Two Finite Dihedral Groups. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 44, pp. 71–81. https://doi.org/10.26516/1997-7670.2023.44.71 (in Russian)
  3. Lysenok I.G. Infinite Burnside groups of even period. Izvestiya RAN. Ser. Matematika, 1996, vol. 60, no. 3, pp. 3–224. (in Russian)
  4. Shlepkin A.A. Groups with a Strongly Embedded Subgroup Saturated with Finite Simple Non-abelian Groups. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 31, pp. 132–141. https://doi.org/10.26516/1997-7670.2020.31.132
  5. Shlepkin A.A., Sabodakh I.V. On Two Properties of Shunkov Groupals. The Bulletin of Irkutsk State University, Series Mathematics, 2021, vol. 35, pp. 103–119.https://doi.org/10.26516/1997-7670.2021.35.103 (in Russian)
  6. Shlepkin A.K. On some periodic groups saturated with finite simple subgroups. Matematicheskie trudy IM SO RAN, 1998, vol. 1, no. 1, pp. 129–138.
  7. Shunkov V.P. On the minimality problem for locally finite groups. Algebra and Logic, 1970, vol. 10, no. 2, pp. 220–248. (in Russian)
  8. Shunkov V.P. On one class of 𝑝-groups Algebra and Logic, 1970, vol. 9, no. 4, pp. 484–496. (in Russian)
  9. Shunkov V.P. On Abelian subgroups in biprimitively finite groups. Algebra and Logic, 1973, vol. 12, no. 5. pp. 603–614. (in Russian)
  10. Ivanov S.V. The free Burnside groups of sufficiently large exponents. Int. J. of Algebra and Computation, 1994, no. 4, pp. 1–308.

Full text (russian)