Permutable subgroups of linear groups

A subgroup H of a group G is said to be permutable in G if HK=KH for every K<=G. This type of subgroups often plays a major role in the context of subnormality, in the study of the subgroup lattice of groups, and in many questions concerning finite and infinite groups. The aim of this paper is to study the general properties of permutable subgroups within the universe of linear groups with respect to three major topics: subnormality conditions, analogs of the Maier-Schmid theorem, and transitivity of permutability. In particular, the behaviour of Zariski closed permutable subgroups is studied in these respects and our main results show for example that the Zariski closure of a permutable subgroup of a linear group is always permutable and subnormal, and that the theorem of Maier-Schmid always hold for Zariski closed subgroups of a linear group.