Normalized root words and reflective power-subgroup completion

Abstract

Fix a prime $p$, and let $P_n(G)$ be the subgroup generated by the $p^n$-th powers in a group $G$. We consider the full category of groups admitting a two-variable word $w$ and an integer $r\geq1$ such that $w(a,b)^p=a^p b^{p^r}$ and $w(a,1)=a$. Every nilpotent group belongs to this category: for each positive class bound $c$, one may take $r=c$ and one word valid in all groups of class at most $c$. For $\widehat G=\varprojlim_n G/P_n(G)$ we prove the exact equality $P_n(\widehat G)=\ker(\widehat G\to G/P_n(G))$. Consequently the inverse-limit and intrinsic power-subgroup topologies coincide, and completion is a reflector onto the complete Hausdorff objects. This gives a categorical setting of the kind requested in Kourovka Problem 10.52.