Lattice characterizations of soluble and supersoluble finite groups
Abstract
Let $G$ be a finite group and ${\cal L}_{sn}(G)$ be the lattice of all subnormal subgroups of $G$. Let $A$ and $N$ be subgroups of $G$ and $G\in {\cal L}$ be a sublattice of ${\cal L}_{sn}(G)$, that is, $A\cap B$, $\langle A, B \rangle \in {\cal L}$ for all $A, B \in {\cal L} \subseteq {\cal L}_{sn}(G)$. Then: $A^{{\cal L}}$ is the $\cal L$-closure of $A$ in $G$, that is, the intersection of all subgroups in $ {\cal L}$ containing $A$ and $A_{{\cal L}}$ is the $\cal L$-core of $A$ in $G$, that is, the subgroup of $A$ generated by all subgroups of $A$ belonging $\cal L$. We say that $A$ is an $N$-${\cal L}$-subgroup of $G$ if either $A\in {\cal L}$ or $A_{{\cal L}} < A < A^{\cal L}$ and $N$ avoids every composition factor $H/K$ of $G$ between $A_{{\cal L}}$ and $ A^{\cal L}$, that is, $N\cap H=N\cap K$. Using this concept, we give new characterizations of soluble and supersoluble finite groups. Some know results are extended.
Keywords
Supporting agencies: The study was supported by the National Natural Science Foundation of China (N. 12101165, 12171126), NNSFC-BRFFR (N. 12311530761), Ministry of Education of the Republic of Belarus (N. 20211778), and the Belarusian Republican Foundation for Fundamental Research (N. F24KI-021)
About the Authors
A.-M. Liu
Hainan University
China
China
School of Mathematics and Statistics
Hainan; Haikou
S. Wang
Hainan University; Tianjin University
China
China
School of Mathematics and Statistics; School of Mathematics
Hainan; Haikou; Tianjin
V. G. Safonov
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Minsk
A. N. Skiba
Institute of Mathematics of the National Academy of Sciences of Belarus; Francisk Skorina Gomel State University
Belarus
Belarus
Minsk; Gomel
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Review
For citations:
Liu A., Wang S., Safonov V.G., Skiba A.N. Lattice characterizations of soluble and supersoluble finite groups. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2024;32(1):17-24.
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