Showing papers on "Torsion-free abelian group published in 2001"

Non-existence of universal members in classes of abelian groups

Abstract: We prove that if µ + < � = cf(�) < µ @0, then there is no universal reduced torsion free abelian group of cardinality �. Similarly if @0 < � < 2 @0. We also prove that if i! < µ + < � = cf(�) < µ @0, then there is no universal reduced separable abelian p-group in �. We also deal with the class of @1-free abelian group. (Note: both results fail if (a) � = � @0 or if (b) � is strong limit, cf(µ) = @0 < µ).

Direct sums of local torsion-free abelian groups

Abstract: The category of local torsion-free abelian groups of finite rank is known to have the cancellation and n-th root properties but not the Krull-Schmidt property. It is shown that 10 is the least rank of a local torsion-free abelian group with two non-equivalent direct sum decompositions into indecomposable summands. This answers a question posed by M.C.R. Butler in the 1960's.

$${\text{IA}}$$ -Automorphisms of Free Products of Two Abelian Torsion-Free Groups

Abstract: Let \(A\) be the free product of two Abelian torsion-free groups, let \(P \triangleleft A\) and \(P \subseteq C\), where \(C\) is the Cartesian subgroup of the group \(A\), and let \(\mathbb{Z}(A/P)\)F contain no zero divisors. In the paper it is proved that, in this case, any automorphism of the group \(A/P'\) is inner. This result generalized the well-known result of Bachmuth, Formanek, and Mochizuki on the automorphisms of groups of the form \(F_2 /R'\), \(R \triangleleft F_2 \), \(R \subseteq F_2 '\), where \(F_2 \) is a free group of rank two.