On the product of two π-decomposable soluble groups

Let the group G = AB be a product of two π-decomposable sub-groups A = Oπ(A) × Oπ' (A) and B = Oπ(B) × Oπ' (B) where π is a set of primes. The authors conjecture that Oπ(A)Oπ(B) = Oπ(B)Oπ(A) if π is a set of odd primes. In this paper it is proved that the conjecture is true if A and B are soluble. A similar result with certain additional restrictions holds in the case 2 ∈ π. Moreover, it is shown that the conjecture holds if Oπ '(A) and Oπ'(B) have coprime orders