ABSTRACT
The question, whether two algebraic structures are isomorphic when their groups of automorphisms are isomorphic, is a fundamental but generally difficult problem in algebra. For vector spaces over fields, or more generally division rings, an affirmative answer was obtained by Rickart [ 5 ] provided that the characteristic is not equal to 2. The corresponding problem for abelian p-groups (p ≥ 5) was solved affirmatively by Leptin [ 3 ] and, more recently, Liebert [ 4 ] has provided a more geometric approach which rederives Leptin’s result and also handles the case of the prime p = 3. His proofs are lengthy and involve an intricate analysis of the situation. In the present note we solve the analogous problem for reduced torsionfree modules over the ring ℤ ^ p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187605/d46d5034-f914-41b4-a96f-9f0731bff786/content/eq840.tif"/> of p-adic integers, where p ≠ 2. Our proof is short, and we believe it is transparent enough to serve as an introduction to the topic. The savings that we achieve are due both to simplification of some existing arguments and also, perhaps surprisingly, to the fact that these torsion-free modules are inherently easier to handle; specifically, since all rank-1 modules are isomorphic, the key homomorphism groups which appear in our proof have a particularly simple form. Some of our simplified methods can be used to rederive Leptin’s result, but they do not, unfortunately, handle the case of p-groups for p = 3. We have not attempted to describe all possible isomorphisms between Aut G and Aut G ˜ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187605/d46d5034-f914-41b4-a96f-9f0731bff786/content/eq841.tif"/> .
