On groups with all proper subgroups finite-by-abelian-by-finite
If \(\mathcal{P}\) is a class of groups it is possible to define a new class \(\overline{\mathcal{P}}\) consisting of all those groups in which every proper subgroup is a \(\mathcal{P}\)-group. Let \(\mathcal{P}=\mathtt{FAF}\) the property of being finite by-abelian-by-finite, hence a group \(G\) is a \(\mathcal{P}\)-group if \(G\) has a subgroup \(K\) with finite index such that the derived subgroup \(K\) is finite. The aim of the paper under review is to study \(\overline{\mathcal{P}}\)-groups. The main result proved in this article is Theorem A: Let \(G\) be a locally graded group. If \(G \in \overline{\mathcal{P}}\), then \(G\) contains a finite normal subgroup \(N\) such that all proper subgroups of \(G/N\) are abelian-by-finite. In particular, if \(G\) is minimal-non-(finite-by-abelian-by-finite), then \(G\) is periodic and finite-by-metabelian. A group \(G\) if a \(\mathtt{CF}\)-group if each subgroup \(H\) of \(G\) is core-finite, i.e. \(H/H_{G}\) is finite. A group \(G\) is a \(\mathtt{CN}\)-group if each subgroup of \(G\) is commensurable to a normal subgroup. Another interesting result is Theorem B: Let \(G\) be a locally graded group whose periodic sections are locally finite. If all proper subgroups of \(G\) are \(\mathtt{CN}\)-groups, then \(G\) contains a finite normal subgroup \(N\) such that all proper subgroups of \(G/N\) are \(\mathtt{CF}\)-groups, hence \(G\) is finite-by-abelian-by-finite.
scientific article; zbMATH DE number 7357040
scientific article