Let the group G = AB be a product of two π-decomposable sub- 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 O π ′ (A) and O π ′ (B) have coprime orders. 2000 Mathematics Subject Classification. 20D20, 20D40. Conjecture. Let π be a

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... of odd primes. Let the group G = AB be the product of two π-decomposable subgroups A = A π × A π ′ and B = B π × B π ′ . Then A π B π = B π A π and this is a Hall π-subgroup of G.