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.

The preliminary research for this paper was done jointly with our colleague Francesco de Giovanni (07.10.1955 - 03.02.2024). He died before the final version was completed. The first author is supported by GNSAGA (INdAM) and is a member of the no profit association "AGTA - Advances in Group Theory and Applications" (www.advgrouptheory.com). Funded by the European Union - Next Generation EU, Missione 4 Componente 1 CUP B53D23009410006, PRIN 2022- 2022PSTWLB - Group Theory and Applications. We thank the referee for their valuable comments and suggestions, which have helped improve the paper.