Generalized soluble $\mathrm{AFM}$-groups

O. Yu. Dashkova

Trudy Instituta matematiki, Tome 21 (2013) no. 1, pp. 52-62

Résumé

We study an $\mathbf{R}\,G$-module $A$ such that $\mathbf{R}$ is an associative ring, $G$ is a group, $C_G(A)=1$ and each proper subgroup $H$ of a group $G$ for which $A/C_A(H)$ is not a minimax $\mathbf{R}$-module, is finitely generated. A group $G$ with these conditions is called an $\mathrm{A}\mathrm{F}\mathrm{M}$-group. It is proved that a locally soluble $\mathrm{A}\mathrm{F}\mathrm{M}$-group $G$ is hyperabelian in the case where $\mathbf{R}=\mathbb{Z}$ is a ring of integers. It is described the structure of an $\mathrm{A}\mathrm{F}\mathrm{M}$-group $G$ in the case where $G$ is a finitely generated soluble group, $\mathbf{R}=\mathbb{Z}$ is a ring of integers and the quotient module $A/C_A(G)$ is not a minimax $\mathbb{Z}$-module.