Separative cancellation and multiple isomorphism in torsion-free Abelian groups
Let all groups be Abelian and torsion-free of finite rank. Recall that a group \(A\) has separative cancellation if \(A\oplus A\cong A\oplus B\cong B\oplus B\) implies \(A\cong B\) and note that the problem of having separative cancellation generalizes the well-known cancellation problem and the \(n\)-th root problem. The group \(A\) is said to have the strong separative cancellation property if \(A\oplus A\cong A\oplus B\) implies \(A\cong B\). Finally recall that two groups \(A\) and \(B\) are multiple isomorphic if \(A^n\cong B^n\) for all \(n\geq 2\) holds. The authors show that for finite rank torsion-free groups separative cancellation, strong separative cancellation and multiple isomorphism are the same. Moreover, using theorems from number theory, they show that examples for the failure of strong separative cancellation are plentiful in the category of torsion-free Abelian groups of finite rank. Finally they show how multiple isomorphism fits into the hierachy of variations of isomorphism for torsion-free Abelian groups of finite rank and they construct in an explicit way families of examples of pairs \((A,B)\) of strongly indecomposable groups which are multiple isomorphic but not isomorphic. The construction relies on a number-theoretic conjecture on odd primes which was verified by computer for almost all primes below \(1000\). Nevertheless, the conjecture itself is still open.
scientific article; zbMATH DE number 1400106
scientific article