Transitivity and full transitivity over subgroups of abelian p-groups

A generalization of Kaplansky's notion of (full-)transitivity is introduced. We say a p-group G is (fully) transitive over a subgroup H, if for all a: is an element of G and y is an element of H with U(x) = U(y) (U(x) less than or equal to U(y)) there exists an automorphism (endomorphism) of G which maps x to y. An adaption of a result by Corner shows that only the elements of infinite height in G are relevant for this definition. Therefore a group G is always transitive and fully transitive over any subgroup which contains only elements of finite height in G. Furthermore we show that a group G is fully transitive over any subgroup with elements of height < lambda, if lambda is the least ordinal for which G/p(lambda)G is not totally projective. We also study subgroups which are maximal with respect to the above definition. These subgroups are dense in the p-adic topology, but not necessarily pure. With the restriction that the length of the group is omega + 1 we can show that these maximal subgroups are indeed pure, and are even N-high subgroups, where N is a subgroup of the first Ulm subgroup. In this setting we also give examples of groups, which contain 2(2N0) non-isomorphic subgroups that are maximal in the above sense.