Научная электронная библиотека ULRICH'S Periodicals Directory

Сибирский государственный университет
науки и технологий имени академика М.Ф. Решетнева

Сибирский аэрокосмический журнал
ISSN 2712-8970 (Print) ISSN 2782-5760 (on-line)

Сведения об образовательной организации

UDK 512.54 Doi: 10.31772/2587-6066-2018-19-3-432-437
MODELING OF THE LAYER STRUCTURE OF INFINTE GROUPS
V. I. Senashov, D. K. Belov*
Institute of Computational Modelling SB RAS 50/44, Akademgorodok, Krasnoyarsk, 660036, Russian Federation Siberian Federal University 79, Svobodny Av., Krasnoyarsk, 660041, Russian Federation *E-mail: white94@inbox.ru
Mathematical modeling of infinite discrete objects is possible if these objects satisfy any conditions of finiteness. If all the layers of elements in the group are finite, a functional description of the power of the layers for such a group is possible. A layer is a set of all elements of the group of the same order. For the first time the infinite layer-finite groups were investigated by S. N. Chernikov initially without a title, and then in his subsequent publications the name of layer-finite groups was fixed. The most intensive studies of the properties of layer-finite groups were carried out in the 1940s and 1950s by S. N. Chernikov, R. Baer, X. X. Muhammedzhan. The paper gives a functional description for some layer-finite groups. It is shown that primary layer-finite groups and layer-finite groups can be very well visualized in the case of two prime divisors of the orders of the elements of the group. For a primary case, it is convenient to use the usual graphical representation. In the case of two prime divisors of the orders of elements of a layer-finite group, visualization of the power functions of the layers by means of surfaces in three-dimensional space is carried out. For a larger number of simple order-divisors, an approach for modeling the layer structure of a complete layer-finite group using subgroup analysis is proposed. In this paper, we study the power functions of the layers for complete layer-finite groups and some finite extensions of these groups, and demonstrate their graphical representations.
Keywords: group, layer, power of layer, order, finite extension.
References

1. Chernikov S. N. Gruppy s zadannymi svoystvami sistemy podgrupp [Groups with given properties of a system of subgroups]. Moscow, Glavnaya redaktsiya fizikomatematicheskoy literatury Publ., 1980, 384 p.

2. Senashov V. I. Sloyno konechnyye gruppy [Layer-finite groups]. Novosibirsk, Nauka Publ., 1993, 158 p.

3. Chernikov S. N. [Infinite layer-finite groups]. Mat. sb. 1948, Vol. 22, No. 1, P. 101–133 (In Russ.).

4. Senashov V. I., Shunkov V. P. [Almost layer finiteness of the periodic part of a group without involutions]. Diskretnaya matematika. 2003, Vol. 15, No. 3, P. 91–104 (In Russ.).

5. Senashov V. I. [Groups with the minimality condition for not almost layer-finite finite subgroups]. Ukr. mat. zhurn. 1991, Vol. 43, No. 7–8, P. 1002–1008 (In Russ.).

6. Senashov V. I. [Sufficient conditions for the almost layer finiteness of the group]. Ukr. mat. zhurn. 1999, Vol. 51, No. 4, P. 472–485 (In Russ.).

7. Senashov V. I. [On groups with strongly embedded subgroups having an almost layer-wise finite periodic part]. Ukr. mat. zhurn. 2012, Vol. 64, No. 3, P. 384–391 (In Russ.).

8. Senashov V. I. Pochti sloyno konechnyye gruppy [Almost layered finite groups]. LAP Lambert Academic Publishing, 2013, 106 p.

9. Senashov V. I. [Pochti sloynaya konechnost' periodicheskoy gruppy bez involyutsiy]. Ukr. mat. zhurn. 1999, Vol. 51, No. 11, P. 1529–1533 (In Russ.).

10. Senashov V. I. [Mutual relations of almost layered finite groups with close classes]. Vestnik SibGAU. 2014, Vol. 15, No. 1, P. 76–79 (In Russ.).

11. Senashov V. I. [Properties of locally cyclic groups]. Siberian Journal of Science and Technology. 2017, Vol. 18, No. 2, P. 290–293 (In Russ.).

12. Senashov V. I. [Graphs of groups]. Materialy IV Vseross. nauch.-metod. konf. s mezhdun. uchastiyem Informatsionnyye tekhnologii v matematike i matematicheskom obrazovanii. Krasnoyarsk, 18–19 November 2015. Krasnoyarsk. state. ped. un-t Publ., P. 93–98 (In Russ.).

13. Senashov V. I. [Layer-finite and almost layerfinite groups]. Informatsionnyye tekhnologii i matematicheskoye modelirovaniye. Izbr. stat'i IX Nauchn. internet-konf. s mezhdun. uchastiyem. 2016, P. 69–87 (In Russ.).

14. Senashov V. I., Oorzhak O. M. O. [Layered graphs of groups]. Vestnik Tuvinskogo gosudarstvennogo universiteta. Tekhnicheskiye i fiziko-matematicheskiye nauki. 2015, Vol. 26, No. 3, P. 145–150 (In Russ.).

15. Senashov V. I., Gerasimova A. M. [On layer graphs of groups]. Aktual’nyye problemy aviatsii i kosmonavtiki. 2017, Vol. 2, No. 13, P. 303–304 (In Russ.).


Senashov Vladimir Ivanovich – Dr. Sc., professor, leading scientific worker, Institute of Computational Modelling SB RAS. E-mail: sen1112home@mail.ru.

Belov Dmitriy Konstantinovich – student of Siberian Federal University. E-mail: white94@inbox.ru.


  MODELING OF THE LAYER STRUCTURE OF INFINTE GROUPS