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A note on mixed \(A\)-reflexive groups. (English) Zbl 1195.20057

For Abelian groups \(A\) and \(G\), the Warfield \(A\)-dual of \(G\), \(G^*=\operatorname{Hom}(G,A)\). There is a canonical homomorphism \(\delta\colon G\to G^{**}\), and \(G\) is called \(A\)-reflexive if \(\delta\) is an isomorphism. The main problem is to characterize, for various choices of \(A\), the \(A\)-reflexive groups. Warfield solved the case for \(A\) rank 1 torsion-free in 1968, and since then, many researchers solved the problem for various classes of torsion-free groups \(A\).
In this paper, the authors deal with mixed groups \(A\) of torsion-free rank 1 and \(p\)-rank \(\leq 1\) for all primes \(p\). Their main theorem is a complete characterization of \(A\)-reflexive groups \(G\) in terms of group theoretic invariants of \(G\). In addition, they find simpler invariants for \(G\) in the special but important case in which \(A\) and \(G\) are self-small.

MSC:

20K21 Mixed groups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
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