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.