The aim of this paper is to prove the following result: Let π be a set of odd primes. If the finite group G = AB is a product of two π-decomposable subgroups A = Oπ(A) × O π (A) and B = Oπ(B) × O π (B), then Oπ(A)Oπ(B) = Oπ(B)Oπ(A) and this is a Hall π-subgroup of G.

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.