Near Isomorphism for Countable-Rank Torsion-Free Abelian Groups

Abstract

The notion of a near isomorphism is extended from finite-rank torsion-free Abelian groups to some classes of infinite-rank groups. The equivalence of different formulations of this notion for a certain class of countable-rank groups is proved.Definition 1.2. Any two torsion-free Abelian groups G and H of finite rank are called nearly isomorphic groups if for any prime p there are monomorphisms Φ p : G → H, Ψ p : H → G such that G/HΨ p and H/GΦ p are finite groups of orders relatively prime to p.Being an equivalence relation on the set of torsion-free Abelian groups of finite rank, it allows one-sided formulations, for example the following. Definition 1.3. Let G and H be torsion-free Abelian groups of finite rank. Then G and H are nearly isomorphic if and only if for any prime p there exists a monomorphism Φ p : G → H such that group H/GΦ p is finite and its order is relatively prime to p.The most complete list of definitions of near isomorphism is contained in [11, Theorem 9.1.4]. For completely decomposable groups of finite rank, near isomorphism coincides with the classical isomorphism, and it is natural to desire the extension of this concept to groups of infinite ranks with the same characteristic. In addition, it is necessary to have such a new near isomorphism definition that could be, in particular, applied to finite-rank groups as near isomorphism in the former sense, since direct summands of groups of infinite rank can have finite ranks.It turned out that all these requirements are satisfied by the following definition (from now on (W ) p denotes the p-primary component of torsion group W ). Definition 1.4 ([7]). Let G and H be torsion-free Abelian groups. Then G and H are called nearly isomorphic groups if for every prime p there exist monomorphisms Φ p :

Solving logistic tasks by parallelizing algorithms of the theory of direct decompositions of torsion-free abelian groups

“…As an appendix in [6], an algorithm is proposed for constructing group X from this class, which has numerous (and not just two) direct decompositions with given sets of ranks of indecomposable terms, which leads to the following statement.…”