Maximal subclasses of local fitting classes

Abstract

A Fitting class F is said to be π-maximal if F is an inclusion maximal subclass of the Fitting class Sπ of all finite soluble π-groups. We prove that F is a π-maximal Fitting class exactly when there is a prime p ∈ π such that the index of the F-radical GF in G is equal to 1 or p for every π-subgroup of G. Hence, there exist maximal subclasses in a local Fitting class. This gives a negative answer to Skiba’s conjecture that there are no maximal Fitting subclasses in a local Fitting class.