Abstract

The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of non-abelian simple groups. The generalized Fitting height of a finite group G is the least number h=h ∗ (G) such that F ∗ h (G)=G , where F ∗ 1 (G)=F ∗ (G) is the generalized Fitting subgroup, and F ∗ i+1 (G) is the inverse image of F ∗ (G/F ∗ i (G)) . It is proved that if a finite group G=AB is factorized by two subgroups of coprime orders, then the nonsoluble length of G is bounded in terms of the generalized Fitting heights of A and B . It is also proved that if, say, B is soluble of derived length d , then the generalized Fitting height of G is bounded in terms of d and the generalized Fitting height of A .