Products of powers and commutators in free pro-p-groups
It is known that an (abstract) group with one defining relation \(r=1\) is not a proper power. For pro-p-groups the analogous statement is false: \textit{D. Gildenhuys} [Invent. Math. 5, 357-366 (1968; Zbl 0159.306)] has constructed examples of pro-p-groups, with a single defining relator that is not a p-th power, that nevertheless contain elements of finite order. The question naturally arises of how special such examples are. It is answered by the following Proposition: Every noncyclic pro-p-group with generators \(x_ i\), \(i\in I\), and single defining relation can be presented by a single defining relation \(r=1\), where r is not a proper power (in the free pro-p-group on \(\{x_ i|\) \(i\in I\})\).
scientific article; zbMATH DE number 4055878