On the extraction of roots in exponential \(A\)-groups.
Let \(A\) be a commutative ring with unity. In exponential \(A\)-group is a group \(G\), equipped with an action by \(A\), such that for all \(g\in G\) and for all \(\alpha\in A\), the element \(g^\alpha\in G\) is uniquely defined and the following axioms hold: (1) \(g^1=g\), \(g^\alpha g^\beta=g^{\alpha+\beta}\), and \((g^\alpha)^\beta=g^{\alpha\beta}\) for all \(g\in G\) and \(\alpha,\beta\in A\). (2) \((h^{-1}gh)^\alpha=h^{-1}g^\alpha h\) for all \(g,h\in G\) and \(\alpha\in A\). (3) If \(g\) and \(h\) are commuting elements of \(G\), then \((gh)^\alpha=g^\alpha h^\alpha\) for all \(\alpha\in A\). There are various examples of exponential \(A\)-groups, including \(A\)-modules, Lyndon's free \(\mathbb Z[x]\)-groups, Baumslag's \(\mathbb Q\)-completion of a free group,\dots. In the article under review the authors investigate the problem of the extraction of roots in exponential \(R\)-groups, where \(R\) is a binomial ring. More exactly, they consider the class \(\mathcal N_R\) of nilpotent \(R\)-powered groups. The main part of the article contains a selection of results on \(\mathcal U_\omega\)-groups (\(\omega\) is some set of primes) and \(\mathcal E_\omega\)-groups in the class \(\mathcal N_R\), where \(\mathcal U_\omega\) denotes the class of groups in which \(p\)-th roots are unique (whenever they exist) for all \(p\in\omega\) and \(\mathcal E_\omega\) denotes the class of groups in which \(p\)-th roots exist for all \(p\in\omega\). In particular, they prove that if \(R\) is a PID and \(G\) is a nilpotent \(R\)-powered group for which root extraction is always possible, then the torsion \(R\)-subgroup of \(G\) lies in the center. Furthermore, if the torsion \(R\)-group is finitely \(R\)-generated, then \(G\) is torsion-free.
scientific article; zbMATH DE number 5880967