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». 2022. Vol 40

On the Existence of 𝑓-local Subgroups in a Group with Finite Involution

Author(s)
Anatoly I. Sozutov1, Mikhail V. Yanchenko1

1Siberian Federal University, Krasnoyarsk, Russian Federation

Abstract
An 𝑓-local subgroup of an infinite group is each its infinite subgroup with a nontrivial locally finite radical. An involution is said to be finite in a group if it generates a finite subgroup with each conjugate involution. An involution is called isolated if it does not commute with any conjugate involution. We study the group 𝐺 with a finite non-isolated involution 𝑖, which includes infinitely many elements of finite order. It is proved that 𝐺 has an 𝑓-local subgroup containing with 𝑖 infinitely many elements of finite order. The proof essentially uses the notion of a commuting graph.
About the Authors

Anatoly I. Sozutov, Dr. Sci. (Phys.–Math.), Prof., Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, sozutov ai@mail.ru

Mikhail V. Yanchenko, Cand. Sci. (Phys.–Math.), Assoc. Prof., Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, yanch1964@yandex.ru

For citation
Sozutov A. I., Yanchenko M. V.On the Existence of 𝑓-local Subgroups in a Group with Finite Involution. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 40, pp. 112–117.

https://doi.org/10.26516/1997-7670.2022.40.112

Keywords
group, 𝑓-local subgroup, finite involution, commuting graph
UDC
518.517
MSC
03C07, 03C60
DOI
https://doi.org/10.26516/1997-7670.2022.40.112
References

1. Belyaev V.V. Groups with almost regular involution. Algebra and Logic, ser. 26, 1987, vol. 5, pp. 315–317. https://doi.org/10.1007/BF01978688

2. Gorenstein D. Finite Simple Groups. An Introduction to Their Classification. New York, Plenum Press, 1982, 333 p.

3. Hall P., Kulatilaka C.R. A property of locally finite groups J. London Math. Soc., 1964, vol. 39, pp. 235–239.

4. Kargapolov M.I., About O.Yu. Shmidt’s Problem. Sib. Math. J., ser. 4, 1963, vol. 1, pp. 232–235.

5. Kurosh A.G. The Theory of Groups. Moscow, Nauka Publ., 1967.

6. Shunkov V.P. On abelian subgroups in biprimitively finite groups. Algebra and Logic, 1973, ser. 12, vol. 5, pp. 603–614.

7. Shunkov V.P. On sufficient criteria for the existence of infinite locally finite subgroups in a group. Albebra and Logic, 1976, ser. 15, vol. 6, pp. 716–737.

8. Shunkov V.P. About embedding primary elements in a group. VO Nauka. Novosibirsk, 1992. (in Russian)

9. Shunkov V.P.𝑇0-groups. Novosibirsk, Nauka Publ., Sibirskaya izdatelskaya firma RAN, 2000, 178 p. (in Russian)

10. Sozutov A.I., Yanchenko M.V. On the existence of 𝑓-local subgroups in a group. Siberian Math. J., 2006, ser. 47, vol. 4, pp. 740–750. https://doi.org/10.1007/s11202-006-0085-7

11. Sozutov A.I., Yanchenko M.V. The 𝑓-local subgroups of the groups with a generalized finite element of order 2 or 4. Siberian Math. J., 2007, vol. 48, no. 5, pp. 923–928.

12. Strunkov N.P., Normalizers and Abelian subgroups of some classes of groups. Izv. AN SSSR. Ser. Math., 1967, ser. 31, vol. 3, pp. 657–670. (in Russian)


Full text (english)