Nilpotent finite groups may be defined by a great number of properties. Of these the following three may be mentioned, since they will play an important part in this investigation. (1) The group is swept out by its ascending central chain (equals its hypercentral). (2) The group is a direct product of p-groups (that is, of its primary components). (3) If S and T are any two subgroups of the group such that T is a subgroup of S and such that there does not exist a subgroup between S and T which is different from both S and T, then T is a normal subgroup of S. These three conditions are equivalent for finite groups; but in general the situation is rather different, since there exists a countable (infinite) group with the following properties: all its elements not equal to 1 are of order a prime number p; it satisfies condition (3); its commutator subgroup is abelian; its central consists of the identity only. A group may be termed soluble, if it may be swept out by an ascending (finite or transfinite) chain of normal subgroups such that the quotient groups of its consecutive terms are abelian groups of finite rank. A group satisfies condition (1) if, and only if, it is soluble and satisfies condition (3) (?2); and a group without elements of infinite order satisfies (1) if, and only if, it is the direct product of soluble p-groups (?3); and these results contain the equivalence of (1), (2) and (3) for finite groups as a trivial special case. If a group without elements of infinite order may be swept out by an ascending chain of subgroups such that each is a normal subgroup of the next one and such that the quotient groups of its consecutive terms are cyclic, then (2) and (3) are equivalent properties, though they no longer imply (1) (?4). If a group satisfies condition (1)-orasuitable weaker conditions-then the elements of finite order in this group generate a subgroup without elements of infinite order which is a direct product of p-groups. A seemingly only slightly stronger condition than (3) is the following property: (3') If S and T are any two subgroups of the group such that T is a subgroup of S and such that there exists at most one subgroup between S and T which is different from both S and T, then T is a normal subgroup of S. Clearly (3') implies (3), though there exist groups which satisfy (3), but not (3'). A closer investigation reveals however that (3') is a much stronger imposition than it seems to be, since it is possible to prove the following theorem: A group, that either does not contain elements of infinite