On the properties of the lattice of τ-closed totally ω-composition formations
https://doi.org/10.29235/1561-2430-2024-60-3-183-194
Abstract
We study the properties of the lattice c τ ω∞ of all τ-closed totally ω-composition formations of finite groups. We prove the modularity of such a lattice of formations for any subgroup functor τ and any nonempty set ω of primes. In particular, we obtain a positive answer to the question of A. N. Skiba and L. A. Shemetkov (2000) about the modularity of the lattice c∞ L of all totally L-composition formations. We establish that the lattice c τ ω∞ is a complete sublattice of the lattice cω ∞ of all totally ω-composition formations of finite groups.
About the Authors
I. P. Los
Belarusian State University; Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Inna P. Los – Postgraduate Student; Junior Researcher
4, Nezavisimosti Ave., 220030, Minsk;
11, Surganov Str., 220072, Minsk
V. G. Safonov
Institute of Mathematics of the National Academy of Sciences of Belarus;Belarusian State University
Belarus
Belarus
Vasily G. Safonov – Dr. Sc. (Physics and Mathematics), Professor, Director; Chief Researcher
11, Surganov Str., 220072, Minsk;
4, Nezavisimosti Ave., 220030, Minsk
References
1. Skiba A. N., Shemetkov L. A. Multiply L composition formations of finite groups. Ukrainian Mathematical Journal, 2000, vol. 52, pp. 898–913. https://doi.org/10.1007/bf02591784
2. Skiba A. N. Algebra of Formations. Minsk, Belaruskaya navuka Publ., 1997. 240 p. (in Russian).
3. Safonov V. G. On modularity of the lattice of totally saturated formations of finite groups. Communications in Algebra, 2007, vol. 35, no. 11, pp. 3495–3502. https://doi.org/10.1080/00927870701509354
4. Safonov V. G., Shemetkov L. A. On sublattices of the lattice of totally saturated formations of finite groups. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2008, vol. 52, no. 4, pp. 34–37 (in Russian).
5. Tsarev A. Inductive lattices of totally composition formations. Revista Colombiana de Matematicas, 2018, vol. 52, no. 2, pp. 161–169. https://doi.org/10.15446/recolma.v52n2.77156
6. Tsarev A. On the lattice of all totally composition formations of finite groups. Ricerche di Matematica, 2019, vol. 68, pp. 693–698. https://doi.org/10.1007/s11587-019-00433-3
7. Shcherbina V. V. On two problems of the theory of partially totally composition formations of finite groups. Prikladnaya matematika & fizika = Applied Mathematics & Physics, 2020, vol. 52, no. 1, pp. 18–32 (in Russian). https://doi.org/10.18413/2687-0959-2020-52-1-18-32
8. Shcherbina V. V. Partially composition formations with a given structure. I. Prikladnaya matematika & fizika = Applied Mathematics & Physics, 2021, vol. 53, no. 3, pp. 171–204 (in Russian). https://doi.org/10.18413/2687-0959-2020-52-1-18-32
9. Los I. P., Safonov V. G. On one-generated and bounded totally ω-composition formations of finite groups. Problemy fiziki, matematiki i tekhniki = Problems of Physics, Mathematics and Technics, 2021, no. 4 (49), pp.101–107 (in Russian). https://doi.org/10.54341/20778708_2021_4_49_101
10. Los I. P., Safonov V. G. Separability of the lattice of τ-closed totally ω-composition formations of finite groups. Trudy Instituta matematiki = Proceedings of the Institute of Mathematics, 2023, vol. 31, no. 2, pp. 44–56 (in Russian).
11. Kamornikov S. F., Shemetkov L. A. On residuals of subnormal subgroups. Algebra i logika = Algebra and Logic, 1995, vol. 34, no. 5, pp. 493–513 (in Russian).
Review
Views: 206
Email this article 