On the behavior of closed subgroups in linear groups

It is well known that any linear group, endowed with the Zariski topology, satisfies the minimal condition on closed subgroups and the maximal condition on closed connected subgroups. The aim of this paper is to study the maximal condition on arbitrary closed (normal) subgroups. Our main results show that every soluble-by-finite linear group G with the maximal condition on closed normal subgroups is topologically finitely generated, and that G even satisfies the maximal condition on closed subgroups, provided that it is also locally nilpotent. Further results are obtained in the case of (affine) algebraic groups and in the case of linear groups with only finitely many closed subgroups. In particular, we describe periodic linear groups that are Zariski-simple, meaning that they have no proper nontrivial closed subgroups.