A Criterion for Modularity of the Subgroup Lattice of a Finite Soluble Group

Let Formula: see text be a finite group and Formula: see text the subgroup lattice of Formula: see text. A subgroup Formula: see text of Formula: see text is called: (i) modular in Formula: see text, if Formula: see text is a modular element (in the sense of Kurosh) of the lattice Formula: see text; (ii) submodular in Formula: see text if Formula: see text has a chain of subgroups Formula: see text, where Formula: see text is modular in Formula: see text for all Formula: see text. If Formula: see text is a subgroup of Formula: see text, then we denote by Formula: see text the subgroup of Formula: see text, generated by all of its subgroups that are modular in Formula: see text. We say that a subgroup Formula: see text is Formula: see text-modular in Formula: see text (Formula: see text), if for some modular subgroup Formula: see text of Formula: see text, containing Formula: see text, Formula: see text avoids the pair Formula: see text, i.e. Formula: see text. We prove that if Formula: see text is a soluble finite group and each of its submodular subgroups is Formula: see text-modular in Formula: see text, where Formula: see text is the nilpotent residual of Formula: see text, then the lattice Formula: see text is modular.