On the σ -Length of Maximal Subgroups of Finite σ -Soluble Groups

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