Groups with all subgroups permutable or of finite rank. (Q424157)

A subgroup \(H\) of a group \(G\) is said to be `permutable' (or `quasinormal') if \(HK=KH\) for all subgroups \(K\) of \(G\). The structure of groups in which all subgroups are permutable (`quasihamiltonian groups') has been completely described by K. Iwasawa; in particular, it turns out that such groups are locally nilpotent and have modular subgroup lattice. In the paper under review, the authors prove that, within a suitable universe of generalized soluble groups (containing all locally (soluble-by-finite) groups), if \(G\) is a group in which all subgroups of infinite rank are permutable, then either \(G\) has finite rank or is quasihamiltonian. This research is motivated by a series of papers dealing with the behavior of groups in which all subgroups of infinite rank have a given property.