Abstract. Finite groups factored to the product of normal subsets are considered. If all factors of composite order are close to groups belonging to some Fitting class then the same holds for the product. In certain cases we can assure that for suitable ordering of the factors the partial products form an ascending chain of normal subgroups.

AMS Subject Classification (1991): 20K01, 20D60

Received September 5, 2000, and in revised form October 25, 2000. (Registered under 2802/2009.)