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An abelian p-group G is called (fully) transitive if for all x, y E G with U G (x) = U G (y) (U G (x) < U G (y)) there exists an automorphism (endomorphism) of G which maps x onto y. It is a long-standing problem of A. L. S. Corner whether there exist non-transitive but fully transitive p-groups with finite first Ulm subgroup. In this paper we restrict ourselves to p-groups of type A, this is to say p-groups satisfying Aut(G) |p ω G = U(End(G) |p ω G). We show that the answer to Corner's question is no if pG is finite and G is of type A.

Fully transitive $p$-groups with finite first Ulm subgroup