A reduction theorem for a conjecture on products of two π-decomposable groups

For a set of primes π, a group X is said to be π-decomposable if X=Xπ×Xπ′ is the direct product of a π-subgroup Xπ and a π′-subgroup Xπ′, where π′ is the complementary of π in the set of all prime numbers. The main result of this paper is a reduction theorem for the following conjecture: “Let π be a set of odd primes. If the finite group G=AB is a product of two π-decomposable subgroups A=Aπ×Aπ′ and B=Bπ×Bπ′, then AπBπ=BπAπ and this is a Hall π-subgroup of G.” We establish that a minimal counterexample to this conjecture is an almost simple group. The conjecture is then achieved in a forthcoming paper.