On permutable strongly 2-maximal and strongly 3-maximal subgroups

We describe the structure of finite solvable non-nilpotent groups in which every two strongly n-maximal subgroups are permutable (n = 2; 3). In particular, it is shown for a solvable non-nilpotent group G that any two strongly 2-maximal subgroups are permutable if and only if G is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable 3-maximal subgroups and with permutable strongly 3-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly 3-maximal subgroups, and we describe 14 classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly n -maximal subgroups if the number of prime divisors of the order of this group strictly exceeds n (n=2; 3).

A CHARACTERIZATION OF SCHMIDT GROUPS BY THEIR ENDOMORPHISM SEMIGROUPS

A Schmidt group is a non-nilpotent finite group in which each proper subgroup is nilpotent. Each Schmidt group G is a solvable group of order ps qv (where p and q are different primes) with a unique Sylow p-subgroup P and a cyclic Sylow q-subgroup Q, and hence G is a semidirect product of P by Q. Denote by [Formula: see text] the class of all Schmidt groups of orders ps qv, where p, q, and v are fixed and s is arbitrary. It is shown in this paper that the class [Formula: see text] can be characterized by the properties of the endomorphism semigroups of the groups of this class. It follows from this characterization that if [Formula: see text] and H is another group such that the endomorphism semigroups of G and H are isomorphic, then [Formula: see text], too.