Groups whose proper subgroups are linear

The aim of this paper is to investigate groups whose proper subgroups are linear. Although there exist locally nilpotent non-linear groups in which every proper subgroup is linear, we prove that, for a large group class containing in particular all hypercentral groups, the linearity of the proper subgroups implies that of the whole group. Moreover, we study groups whose proper subgroups are linear of characteristic 0, showing in particular that these groups are themselves linear of characteristic 0 provided that they are either locally nilpotent or locally finite.