Abstract
Let sigma = {sigma(i): i is an element of I} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is sigma-primary if all the prime factors of vertical bar G vertical bar belong to the same member of sigma. G is said to be sigma-soluble if every chief factor of G is sigma-primary, and G is sigma-nilpotent if it is a direct product of sigma-primary groups. It is known that G has a largest normal sigma-nilpotent subgroup which is denoted by F sigma(G). Let n be a non-negative integer. The n-term of the sigma-Fitting series of G is defined inductively by F-0(G)=1, and Fn+1(G)/Fn(G)=F sigma(G/Fn(G)). If G is sigma-soluble, there exists a smallest n such that Fn(G)=G. This number n is called the sigma-nilpotent length of G and it is denoted by l(sigma)(G). If F is a subgroup-closed saturated formation, we define the sigma-F-length n sigma(G,F) of G as the sigma-nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a sigma-soluble, then n(sigma)(A, F) = n(sigma)(G,F) - i for some i is an element of{0,1,2}.