Finite groups with permuted strongly generalized maximal subgroups

Let G be a finite group. If there is a maximal subgroup M in G such that H <= Mand H is a maximal subgroup of M, then H is called the 2-maximal subgroup of G. The 3-maximal subgroups can be defined similarly. Note that the n-maximal subgroup of the group G is called strongly the n-maximal if it is not the n-maximal subgroup in any proper subgroup of the group G. This paper is devoted to describing the structure of the groups in which any strongly 2-maximal subgroup is permutable with the arbitrary strongly 3-maximal subgroup. The class of groups with this property is proved to coincide with the class of groups in which any 2- maximal subgroup is permuted with the arbitrary 3-maximal subgroup, and, as a consequence, such groups are solvable. As an auxiliary result, this work presents a description of groups in which any strongly 2-maximal subgroup is permutable with an arbitrary maximal subgroup. The class of such groups is shown to coincide with the class of the groups in which any 2-maximal subgroup is permutable with all maximal subgroups and, as a consequence, such groups are supersolvable.