Let $G$ be a finite group, and let ${\cal L}(G)$ be the lattice of all subgroups of $G$. A subgroup $M$ of $G$ is called modular in $G$ if $M$ is a modular element (in the Kurosh sense) of the lattice ${ \cal L}(G)$, i.e., if (1) $\langle X, M \cap Z \rangle=\langle X, M \rangle \cap Z$ for all $X \leq G, Z \leq G$ such that $X \leq Z$, and (2) $\langle M, Y \cap Z \rangle=\langle M, Y \rangle \cap Z$ for all $Y \leq G, Z \leq G$ such that $M \leq Z$. If $A$ is a subgroup of $G$, then $A_{m G}$ is the subgroup of $A$ generated by all its subgroups that are modular in $G$. We say that a subgroup $A$ is $N$-modular in $G$ ($N\leq G$) if, for some modular subgroup $T$ of $G$ containing $A$, $N$ avoids the pair $(T, A_{mG})$, i.e. $N\cap T=N\cap A_{mG}$. Using these notions, we give new characterizations of $p$-soluble and $p$-supersoluble finite groups.
Keywords: finite group, $p$-soluble group, $p$-supersoluble group, modular subgroup, $N$-modular subgroup
Received May 13, 2024
Revised June 12, 2024
Accepted June 17, 2024
Funding Agency: This work was supported by the National Natural Science Foundation of China (project nos. 12101165 and 12171126) as well as jointly by the National Natural Science Foundation of China and the Belarusian Republican Foundation for Fundamental Research (project no. 12311530761). The research of V.G. Safonov and A.N. Skiba was supported by the Belarusian Republican Foundation for Fundamental Research (project no. F24KI-021).
A-Ming Liu, Cand. Sci., (Phys.-Math.), Assistant professor, School of Mathematics and Statistics, Hainan University, Haikou, Hainan, 570228 P.R. China, e-mail: amliu@hainanu.edu.cn
Sizhe Wang, graduate student, School of Mathematics and Statistics, Hainan University, Haikou, Hainan, 570228 P.R. China; School of Mathematics, Tianjin University, Tianjin, 300072 China, e-mail: 169518909@QQ.com
Vasily Grigorievich Safonov, Dr. Sci., (Phys.-Math.), Professor, Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, 220072 Belarus, e-mail: vgsafonov@im.bas-net.by
Alexander Nikolaevich Skiba, Dr. Sci., (Phys.-Math.), Professor, Department of Mathematics and Programming Technologies, Francisk Skorina Gomel State University, Gomel, 246019 Belarus; Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, 220072 Belarus, e-mail: alexander.skiba49@gmail.com
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Cite this article as: A-Ming Liu, Sizhe Wang, V.G. Safonov, A.N. Skiba. Lattice characterizations of p-soluble and p-supersoluble finite groups. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 180–187.
