Groups in which each subnormal subgroup is commensurable with some normal subgroup
Two subgroups \(H, K\) of a group \(G\) are commensurable, if \(|H:H\cap K|\) and \(|K:H\cap K|\) are both finite. The subgroup \(H\) is called a cn-subgroup of \(G\) if there is a normal subgroup \(N\) of \(G\) such that \(H\) is commensurable with \(N\), or equivalently \(|HN:H\cap N|\) is finite. The subgroup \(H\) is a cf-subgroup (for core-finite) if \(|H:H_G|\) is finite and \(H\) is an nn-subgroup (for nearly normal) if \(|H^G:H|\) is finite. It is clear that nn-subgroups and cf-subgroups are examples of cn-subgroups. The groups in which every subgroup is an nn-subgroup are precisely the finite-by-abelian groups, as shown in a famous paper of [\textit{B. H. Neumann}, Math. Z. 63, 76--96 (1955; Zbl 0064.25201)]. A group \(G\) in which each subgroup is core-finite is called a CF-group and a group in which each subgroup is cn is called a CN-group. In the paper under review, the authors consider the class T[*] of groups in which the subnormal subgroups are cn. Thus T[*] is the class of groups in which the relation of being commensurable with a normal subgroup is transitive. The authors prove, among other things: Theorem A: Let \(G\) be a subsoluble T[*]-group. Then \(G\) is finite-by-metabelian and hence \(G\) is soluble. Furthermore, all finitely generated subgroups of \(G\) are CF-groups, so \(G\) is locally (abelian-by-finite). Theorem B: Let \(G\) be a locally nilpotent T[*]-group. Then \(G\) is hypercentral. Furthermore, if \(G\) is periodic, then \(G\) is a CN-group with a nilpotent subgroup of index at most \(2\). They show also that for finitely generated soluble groups the conditions CF, CN,T[*] are equivalent to two further conditions \(T^*\) and \(T_*\) which force the group to be abelian-by-finite. The groups with property \(T_*\) were first studied in the paper of \textit{S. Franciosi} et al. [Commun. Algebra 23, No. 14, 5483--5497 (1995; Zbl 0839.20039)] (these are the groups whose subnormal subgroups are core-finite). The class \(T^*\) was studied by \textit{C. Casolo} [Rend. Semin. Mat. Univ. Padova 81, 107--149 (1989; Zbl 0692.20028)] consisting of groups whose subnormal subgroups are nearly normal. This is a very nice, very appropriate paper dedicated to two fine mathematicians, Carlo Casolo and Silvana Rinauro.
scientific article; zbMATH DE number 7369594
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