ADJUNCTION OF ROOTS TO PROFINITE GROUPS

Abstract

In this paper, when $G'$ is a group obtained by adjoining a $n$th-root of $g$ to a given group $G$, where $n$ is a nonzero natural number and $g$ is an element of $G$ of infinite order, we compute the profinite completion $\widehat{G'}$ of $G'$. Also, given $G$ a profinite group in which any subgroup of finite index is open, $n$ a nonzero natural number, $g$ an element of $G$, and $x$ an element not belonging to $G$, we point out necessary and sufficient condition under which the group obtained by adjoining roots to the profinite group $G$ remains again profinite. Our proofs make use of theoretico-combinatorial methods.