Relative injectivity and equivalence theorems (Q2725478)

Two subgroups \(H\) and \(K\) of an Abelian group \(G\) are called equivalent if there is an automorphism of \(G\) sending \(H\) onto \(K\). The authors investigate the relationships between several notions related to equivalence, all in the category of torsion-free Abelian groups of finite rank.NEWLINENEWLINENEWLINEThe relevant definitions are listed first. A group \(G\) (always torsion-free of finite rank) is strongly homogeneous if for any two pure rank-1 subgroups \(X\) and \(Y\) of \(G\) there is an automorphism of \(G\) sending \(X\) onto \(Y\). The group \(G\) is quasi-pure injective (qpi) if any homomorphism from a pure subgroup into \(G\) lifts to an endomorphism of \(G\). The group \(G\) has minimal test for the equivalence of pure subgroups (mteps) if whenever \(H\) and \(K\) are isomorphic pure subgroups of \(G\), \(H\) and \(K\) are equivalent in \(G\). Finally, \(G\) has the lifting isomorphism property for pure subgroups (lips) if whenever \(H\) and \(K\) are pure subgroups of \(G\) and \(f\colon H\to K\) is an isomorphism, there is an automorphism of \(G\) that lifts \(f\).NEWLINENEWLINENEWLINEIn restricted settings, all these notions coincide, as the following shows.NEWLINENEWLINENEWLINECorollary 1.1. Let \(G\) be indecomposable and homogeneous such that \(\{p\text{ prime}:pG=G\}\) is finite. The following are equivalent: (1) \(G\) is strongly homogeneous; (2) \(G\) is qpi; (3) \(G\) has lips; (4) \(G\) has mteps.NEWLINENEWLINENEWLINEIn the homogeneous case, mteps and qpi are equivalent with a miniscule exception.NEWLINENEWLINENEWLINECorollary 2.9. Let \(G\) be homogeneous. (1) If \(G\) is qpi, then \(G\) has mteps; (2) If \(G\) has mteps and \(G\) is not a rank-3 group for which \(\{p\text{ prime}:pG=G\}\) is infinite, then \(G\) is qpi.NEWLINENEWLINENEWLINEThe rank-3 condition in (2) of the corollary can not be removed -- the authors provide an example of a rank-3 group having mteps that is not qpi.NEWLINENEWLINENEWLINEA weakened version of the lips condition is useful in the nonhomogeneous case. A group \(G\) has the weak lips condition if any two isomorphic rank-1 pure subgroups are equivalent. Let \(R\) be a subring of an algebraic number field such that for every \(r\) in \(R\) there is a prime \(p\) such that \(pR\neq R\) and \(r-k\in pR\) for some integer \(k\). Then \(R\) is called a pointwise locally rational (plr) ring.NEWLINENEWLINENEWLINETheorem 4.5. Let \(G\) be strongly indecomposable and nonhomogeneous. Then (1) \(G\) has lips if and only if \(G\) has weak lips and \(\text{End}(G)\) is a plr ring. (2) Suppose that either the rank of \(\text{End}(G)\) is not 3 or \(pG=G\) for only finitely many primes. Then \(G\) has mteps if and only if \(G\) has lips.