Finite groups with permuted strongly generalized maximal subgroups

Abstract

The structure of finite groups in which any strictly 2-maximal subgroup permutes with an arbitrary strictly 3-maximal subgroup is described. It is shown that the class of groups with this property coincides with the class of groups in which any 2-maximal subgroup permutes with an arbitrary 3-maximal subgroup, and, as a consequence, such groups are solvable. As auxiliary results, we describe the structure of groups in which any strictly 2-maximal subgroup permutes with an arbitrary maximal subgroup. In particular, it is shown that the class of such groups coincides with the class of groups in which any 2-maximal subgroup commutes with all maximal subgroups, and, as a consequence, such groups are supersoluble.