Groups Whose Proper Subgroups Have a Finite-Index Subgroup with Modular Subgroup Lattice

If X is a class of groups, a group is minimal non-X if it is not an X-group, but all its proper subgroups belong to X. We prove here that, at least within the universe of non-perfect periodic locally graded groups, the properties of being minimal non-MF and minimal non-(abelian-by-finite) coincide, where MF is the class of groups having a (normal) subgroup with a modular subgroup lattice of finite index. Moreover, we investigate the behaviour of uncountable periodic groups of regular cardinality ℵ in which all proper subgroups of cardinality ℵ belong to MF. It is proved here that such a group G contains a finite-index subgroup having a modular subgroup lattice, provided that G′≠G. A corresponding result for groups whose proper subgroups of large cardinality contain a quasihamiltonian subgroup of finite index is also proved. Recall that a group G is called quasihamiltonian if XY=YX for all subgroups X and Y of G.