Finite groups all of whose 2-maximal subgroups are π-decomposable

Abstract

Let π be an arbitrary set of primes. A very broad generalization of the notion of nilpotent group is the notion of π-decomposable group, which is the direct product of a π-group and a π′-group. In this paper, we obtain a description of finite π-indecomposable groups all of whose 2-maximal subgroups are π-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.