Groups with all subgroups either modular or soluble of finite rank
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.
scientific article; zbMATH DE number 7290696
scientific article