Countable mixed Abelian groups with very nice full-rank subgroups
A group G is a Warfield group if it contains a free valuated subgroup with basis X such that \(<X>\) is nice in G and \(G/<X>\) is a totally projective torsion group. Warfield groups are classified by invariants, and they form a tractable and well-behaved class of groups. The authors consider a ''nearby'' class of groups G: N is allowed to be any countable torsion-free subgroup of G while G/N is required to be a direct sum of cyclic p-groups. In contrast to the class of Warfield groups this new class of groups is shown to be ''wild'' and unclassifiable. This is done by proving a realization theorem of algebras in the endomorphism algebra of such groups: Let A be a p-reduced torsion-free countable Z-algebra. By the well-known theorem of Corner there is a countable torsion-free group N such that \(End(N)=A\). An extension G of N is constructed such that G/N is a countable direct sum of cyclic p-groups, the torsion part tG is a non-zero direct sum of cyclic p-groups and \(End(G)=A\oplus Hom(G,tG)\). In particular, the groups G lie in the new class described above, and the diversity of possible endomorphism rings shows that the class is wild and untractable. Furthermore, given A, there are \(2^{\aleph_ 0}\) groups G with the same subgroup N, the same torsion subgroup T, the same quotient G/T and with \(End(G)=A\oplus Hom(G,T)\). If G and G' are two distinct groups of this set then \(Hom(G,G')=Hom(G,T).\) \(\{\) The authors are rather careless about stating hypotheses and about their notation. The following are indicative. In (2.4) N must contain A as assured by Corner's Theorem on the same page. The latter should also include the statement that \(G_ i\) is S-pure in A. The sequences introduced just before (2.4) are all assumed to be monotone null- sequences (in particular \(\tau \sigma^{-1})\). The element b on p. 5 should be \(b=\sum s_ n\pi_ n\). The elements \(e_ i\) making their unannounced appearance on p. 6 should be interpreted as \(\pi_ i.\}\)
scientific article; zbMATH DE number 3925119
scientific article