A nontransitive, fully transitive primary group

Abstract

AU groups in this note are additively written p-primary abelian groups. If G is such a group, WC define the subgroup p?G, for all ordinals cy, in the usual inductive manner. If G is reduced and x is a nonzero clement of G, then we define h,(x) to be the first ordinal a such that x E p”G and x $ pULIG. We set h,(O) == c/3 and follow the convention CL < co for all ordinals a. \Yith each element s E G we associate its Uht sequence u,(x) = (a,, , aI ,..., a,& ,...) where LY ,, : h,(p%). Um sequences are ordered in the obvious pointwise manner. Following Kaplansky [5], we call a reduced primary group G transitire if for each pair of elements X, y E G such that U,(X) = UC;(y) there exists an automorphism of G mapping x to y and fzdly transitive if UG(x) < V,(y) implies the existence of an endomorphism of G mapping x to y. In [J] it is shown that transitivity implies full transitivity at least when p f 2. The existence of reduced primary groups that are neither transitive nor fully transitive was first established in [6]. -4s indicated by our title, we shall here construct a reduced primary group that is fully transitive but not transitive. We rely heavily on Corner’s construction in [1] of primary groups with certain prescribed endomorphism rings. Unfortunately, Corner’s methods, which are based on ideas originally due to Crawlcy [2], treat only primary groups without elements of infinite height, that is, groups G for which pwG = 0. Hut, of course, groups without elements of infinite height are both transitive and fully transitive. Rather than generalizing Corner’s results to a wider class of groups, we shall use a make-shift construction that yields a group G with p”G cyclic and with appropriately restricted endomorphism ring. An endomorphism C#J of G will be called a small edomorphism if for each positive integer k there exists a nonnegative integer n such that (p”G)[p”] is contained in the kernel of 4. We denote the endomorphism ring of G as End G and the ideal of small endomorphisms