On an open problem in the theory of modular subgroups

  • Liu Aming-Ming Hainan University, 58 Renmin Avenue, Haikou 570228, Hainan Province, China
  • Guo Wenbin Hainan University, 58 Renmin Avenue, Haikou 570228, Hainan Province, China
  • Inna N. Safonova Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus
  • Alexander N. Skiba Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus
DOI: https://doi.org/10.33581/2520-6508-2023-2-28-34

Abstract

Let $G$ be a finite group. Then a subgroup $A$ of group $G$ is said to be modular in $G$ if (i) $\langle X, A \cap Z \rangle=\langle X, A \rangle \cap Z$ for all $X \leq G, Z \leq G$ such that $X \leq Z,$ and (ii) $\langle A, Y \cap Z \rangle=\langle A, Y \rangle \cap Z$ for all $Y \leq G, Z \leq G$ such that $A \leq Z.$ We obtain a description of finite groups in which modularity is a transitive relation, that is, if A is a modular subgroup of K and K is a modular subgroup of G, then A is a modular subgroup of G. The result obtained is a solution to one of the old problems in the theory of modular subgroups, which goes back to the works of A. Frigerio (1974), I. Zimmermann (1989).

Author Biographies

Liu Aming-Ming, Hainan University, 58 Renmin Avenue, Haikou 570228, Hainan Province, China

associate professor at the School of Science

Guo Wenbin, Hainan University, 58 Renmin Avenue, Haikou 570228, Hainan Province, China

professor at the School of Science

Inna N. Safonova, Belarusian State University, 4 Niezaliezhnasci Avenue, Minsk 220030, Belarus

PhD (physics and mathematics), docent; deputy dean of the faculty of applied mathematics and computer science for research

Alexander N. Skiba, Francisk Skorina Gomel State University, 104 Savieckaja Street, Gomiel 246028, Belarus

doctor of science (physics and mathematics), full professor; professor at the department of algebra and geometry, faculty of mathematics and programming technologies

References

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Published
2023-07-24
Keywords: finite group, modular subgroup, submodular subgroup, M-group, Robinson complex
Supporting Agencies This work was supported by the National Natural Science Foundation of China (grant No. 12171126) and the Natural Science Foundation of Hainan Province of China (grant No. 621RC510). The research of the third author was supported by the Ministry of Education of the Republic of Belarus (state registration No. 20211328).
How to Cite
Aming-Ming, L., Wenbin, G., Safonova, I. N., & Skiba, A. N. (2023). On an open problem in the theory of modular subgroups. Journal of the Belarusian State University. Mathematics and Informatics, 2, 28-34. https://doi.org/10.33581/2520-6508-2023-2-28-34
Issue
No 2 (2023): Journal of the Belarusian State University. Mathematics and Informatics
Section
Mathematical Logic, Algebra and Number Theory

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