Maximal fitting subclasses of the class of all finite π-groups

Abstract

Call a Fitting class \(\mathfrak{F}\) π-maximal if \(\mathfrak{F}\) is (inclusion-)maximal in the class \(\mathfrak{C}_\pi\) of all finite π-groups, where π stands for a nonempty set of primes. We establish a π-maximality criterion for a Fitting class \(\mathfrak{F}\) of finite π-groups: we prove that a nontrivial Fitting class \(\mathfrak{F}\) is π-maximal if and only if there is a prime p ∈ π such that, for every π-group G, the index of the \(\mathfrak{F}\)-radical \(G_\mathfrak{F}\) in G is equal to 1 or p. This implies Laue’s familiar result on a necessary and sufficient condition of the maximality of an arbitrary Fitting class of finite groups in the class \(\mathfrak{C}\) of all finite groups. The π-maximality criterion obtained also gives a confirmation of the negative solution of Skiba’s Problem asking whether a local Fitting class has no inclusion-maximal Fitting subclasses (see Problem 13.50, The Kourovka Notebook: Unsolved Problems in Group Theory, 14th ed., Sobolev Institute of Mathematics, Novosibirsk, 1999).