Groups with all subgroups either modular or soluble of finite rank
type: Item
instance of: scholarly article
title: Groups with all subgroups either modular or soluble of finite rank
zbMATH Open document ID: 1462.20012
author: Fausto De Mari
published in: Journal of Algebra
publication date: 2021-01-06T00:00:00Z
review text: This paper is devoted to a study of the effect on a group $G$ of imposing restrictions on the non-modular subgroups of $G$, where a subgroup $H$ of $G$ is called modular if $\langle H,X\rangle\cap Y=\langle X,H\cap Y\rangle$ whenever $X\leq Y\leq G$ and $\langle H, X\rangle\cap Y = \langle H, X\cap Y\rangle$ whenever $H\leq Y\leq G$. The groups $G$ are also restricted to a very wide class of groups designed to avoid difficult groups like the Tarski monsters. Thus let $G$ be a infinite group such that every non-trivial finitely generated subgoup of $G$ has a non-trivial finite image and assume that every non-modular subgroup of $G$ lies in some class $\mathbf{X}$. Theorem 8 proves that $\mathbf{X}$ can be taken to be the class of soluble groups with finite abelian sectional rank (the class FAR in the Lennox-Robinson classification system) if and only if the subgroup lattice of $G$ is modular, or $G$ is soluble with finite abelian sectional rank, or $G''$ is a finite perfect minimal non-soluble group, the subgroup lattice of $G/G''$ is modular and $G$ has finite abelian sectional rank. Using Theorem 8 one can easily derive the corresponding result for $\mathbf{X}$ the class of soluble groups of finite rank (Corollary 9), for $\mathbf{X}$ the class $\mathbf{S}_1$, that is the class FATR in Lennox-Robinson system, (Corollary 10), for $\mathbf{X}$ the class of soluble minimax groups (Corollary 11) and reduce the case where $\mathbf{X}$ is the class of polycyclic groups to a study of certain soluble minimax groups (Corollary 12). The polycyclic group case is completed in the long and complicated Section 4, the outcome being Theorem 23, itself too long to state here. The case where $\mathbf{X}$ is the class of finite groups, which does not quite fit the general pattern above, is settled in the author's Theorem 4. Finally, the author's paper begins with a very thorough survey of past work in this area.
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 7290696
zbMATH Keywords: locally graded group
zbMATH Keywords: modular subgroup lattice
zbMATH Keywords: quasihamiltonian group
zbMATH Keywords: polycyclic
zbMATH Keywords: minimax
zbMATH Keywords: finite rank
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2020.12.005
OpenAlex ID: W3112519357
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DOI: 10.1016/J.JALGEBRA.2020.12.005
description: scientific article; zbMATH DE number 7290696