Transitivity and full transitivity for p-local modules

A p-local module M is called (fully) transitive if for all x,y ? Mx,y\in M with U_M(x) = U_M(y) ( U_M(x)\leqq U_M(y)U_M(x)\leqq U_M(y)) there exists an automorphism (endomorphism) of M which maps x onto y. In this paper we examine the relationship of these two notions in the case of p-local modules. We show that a module M is fully transitive if and only if M?MM\oplus M is transitive in the case where the divisible part of M/tMM/tM has rank at most one. Moreover, we show that for the same class of modules transitivity implies full transitivity if p > 2. This extends theorems of Files, Goldsmith and of Kaplansky for torsion p-local modules.