Maximal subclasses of local fitting classes

Abstract

A Fitting class $\mathfrak{F}$ is said to be π-maximal if $\mathfrak{F}$ is an inclusion maximal subclass of the Fitting class $\mathfrak{S}_\pi$ of all finite soluble π-groups. We prove that $\mathfrak{F}$ is a π-maximal Fitting class exactly when there is a prime p ∊ π such that the index of the $\mathfrak{F}$-radical $G_\mathfrak{F}$ 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 (see [1, Question 13.50]).