On torsion in metabelian pro-p-groups with one defining relation (Q798430)

Theorem. Let F be the free metabelian pro-p-group freely generated by \(x_ 1,...,x_ n\), let X be the normal closure of \(x_ 1\), and let R be generated by \(r=x_ 1^{p^{\ell}}c\), where \(c\in F'\) and \(\ell\geq 1\). Let \(\alpha:{\mathbb{Z}}_ p\to {\mathbb{Z}}/p{\mathbb{Z}}\) be the natural homomorphism and \(d_ i\) the partial derivative with respect to \(x_ i\) of \({\mathbb{Z}}_ p[[F]]\) in Z'\({}_ p[[F/F'X]]\). Then F/R is torsion-free if and only if there is an i such that \(\alpha d_ i(r)\neq 0\).