On the product of a π-group and a π-decomposable group

The main result in the paper states the following: Let π be a set of odd primes. Let the finite group G=AB be the product of a π-decomposable subgroup A=Oπ(A)×Oπ′(A) and a π-subgroup B. Then Oπ(A)⩽Oπ(G); equivalently the group G possesses Hall π-subgroups. In this case Oπ(A)B is a Hall π-subgroup of G. This result extends previous results of Berkovich (1966), Rowley (1977), Arad and Chillag (1981) and Kazarin (1980) where stronger hypotheses on the factors A and B of the group G were being considered. The results under consideration in the paper provide in particular criteria for the existence of non-trivial soluble normal subgroups for a factorized group G.