Finite Groups All of Whose 2-Maximal Subgroups Are π-Decomposable

Let pi be an arbitrary set of primes. A very broad generalization of the notion of nilpotent group is the notion of pi-decomposable group, which is the direct product of a pi-group and a pi '-group. In this paper, we obtain a description of finite pi-indecomposable groups all of whose 2-maximal subgroups are pi-decomposable. The proof involves the author's recent results connected with the notion of control of the prime spectrum of a finite simple group. Finite nonnilpotent groups all of whose 2-maximal subgroups are nilpotent were studied by Z. Janko in 1962 in the case of nonsolvable groups and by the author in 1968 in the case of solvable groups.