A criterion for the existence of normal π-complements in finite groups

Abstract

The purpose of this article is to prove, using the classification of the finite simple groups, the following conjecture: Let π be a set of odd primes, then a finite group is π-homogeneous if and only if it is π′-closed. Using this, several open problems can be settled, including an affirmative answer to the following problem of Baer: Let G be a finite group and π ⊆ π( G). Suppose that G is both π-homogeneous and π′-homogeneous. Is G a direct product of a π-group and a π′-group? Finally, we note that the proof of the conjecture yields proof to some theorems proved earlier without using the classification of the finite simple groups.