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

Abstract

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 = A B is a product of two π-decomposable subgroups A = Aπ × Aπ and B = Bπ × Bπ , then Aπ Bπ = Bπ Aπ 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. © 20