Adjunction of Roots to Profinite Groups

In this paper, when $G'$ is a group obtained by adjoining a nth-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 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.