Normality and modularity conditions on subgroups.

A group G is said to have finite Prà¼fer rank r if every finitely generated subgroup of G can be generated by at most r elements, and r is the least positive integer with such property; if such an r does not exist, we will say that the group G has infinite rank. (Generalized) soluble groups of infinite rank in which all subgroups of infinite rank are either normal or self-normalizing and groups in which all subgroups of infinite rank are either normal or contranormal have been considered. In both cases it has been proved that subgroups of finite rank have the same property satisfied by subgroups of infinite rank. The lattice-theoretic interpretation of normality is modularity. It has been proved that if G is a finitely generated soluble group such that every infinite set of cyclic subgroups contains two subgroups H and K which are modular in <H,K>, then G is central-by-finite. Finally we can remark that permutability has some generalizations. In particular we say that a subgroup H is nearly permutable if there exists a permutable subgroup K of G containing H such that the index |K:H| is finite. Generalized radical groups of infinite rank in which all subgroups of infinite rank are nearly permutable have been considered. First of all it has been proved that the commutator subgroup G' of G is locally finite and then it has proved, in non-periodic case, that either G is an FC-group or G/T(G) is a torsion-free abelian group with rank 1.