2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/132518We characterize thorn-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets.Logic03C10, 03C45, 03C64text