On a sublattice of the lattice of normal Fitting classes

Abstract:

Let ${\mathbf {L}}$ be the set of all Fitting classes $\mathfrak {F}$ with the following two properties: (i) $\mathfrak {F} \supseteq \mathfrak {N}$, the class of all finite nilpotent groups, and (ii) every $\mathfrak {F}$-avoided, complemented chief factor of any finite soluble group $G$ is partially $\mathfrak {F}$-complemented in $G$. It is shown that ${\mathbf {L}}$ is a complete sublattice of the complete lattice ${\mathbf {N}}$ of all nontrivial normal Fitting classes, and, moreover, it is lattice isomorphic to the subgroup lattice of the Frattini factor group of a certain abelian torsion group due to H. Lausch.