On the product of two π-decomposable groups

[EN] The aim of this paper is to prove the following result: let π be a set of odd primes. If the finite group G = AB is a product of two π-decomposable subgroups A = Oπ(A)×Oπ (A) and B = Oπ(B)×Oπ (B), then Oπ(A)Oπ(B)=Oπ(B)Oπ(A) and this is a Hall π-subgroup of G.
The second and third authors were supported by Proyecto MTM2010-19938-C03-02, Ministerio de Economia y Competitividad, Spain. The first author would like to thank the Universitat de Valencia and the Universitat Politecnica de Valencia for their warm hospitality during the preparation of this paper and the RFBR Project 13-01-00469 for its support. The authors are also grateful to A. S. Kondratiev for helpful comments and suggestions during his visit to Valencia.