A duality proof of a theorem of P. Hill

Abstract

A duality argument is given to prove the equivalence of a recent theorem of P. Hill and the main step in Zippin's proof of Ulm's theorem. Let G and H be isomorphic finite abelian p-groups with isomorphic sub- groups A and B, respectively, and set X = G/A, Y = H/B. In his 1935 proof of Ulm's theorem, Zippin established the following result. Theorem (Zippin (2, p. 30), (3)). If 0: A B is an isomorphism, then there exists an isomorphism G -- H that induces o if and only if f# preserves heights.