ON THE PRODUCT OF TWO π-DECOMPOSABLE SOLUBLE GROUPS

Let the group G = AB be a product of two pi-decomposable subgroups A = O-pi(A) x O-pi'(A) and B = O-pi(B) x O-pi'(B) where pi is a set of primes. The authors conjecture that O-pi(A)O-pi(B) = O-pi(B)O-pi(A) if pi 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 is an element of pi. Moreover, it is shown that the conjecture holds if O-pi'(A) and O-pi'(B) have coprime orders.