If \({\mathfrak {X}}\) is a class of groups, a group is minimal non- \({\mathfrak {X}}\) if it is not an \({\mathfrak {X}}\) -group, but all its proper subgroups belong to \({\mathfrak {X}}\) . We prove here that, at least within the universe of non-perfect periodic locally graded groups, the properties of being minimal non- \({\mathfrak {M}}{\mathfrak {F}}\) and minimal non-(abelian-by-finite) coincide, where \({\mathfrak {M}}{\mathfrak {F}}\) 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 \(\aleph \) in which all proper subgroups of cardinality \(\aleph \) belong to \({\mathfrak {M}}{\mathfrak {F}}\) . It is proved here that such a group G contains a finite-index subgroup having a modular subgroup lattice, provided that \(G' \ne G\) . 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.
