Lattice-theoretic characterizations of the group classes ${\mathfrak{N}^{k-1}\mathfrak{A}}$ and ${\mathfrak{N}^k}$ for k ≥ 2

Abstract

Let ${2\leq k\in \mathbb{N}}$. Recently, Costantini and Zacher obtained a lattice-theoretic characterization of the classes ${\mathfrak{N}^k}$ of finite soluble groups with nilpotent length at most k. It is the aim of this paper to give a lattice-theoretic characterization of the classes ${\mathfrak{N}^{k-1}\mathfrak{A}}$ of finite groups with commutator subgroup in ${\mathfrak{N}^{k-1}}$; in addition, our method also yields a new characterization of the classes ${\mathfrak{N}^k}$. The main idea of our approach is to use two well-known theorems of Gaschütz on the Frattini and Fitting subgroups of finite groups.