On the Group of Continuous Automorphisms of Some Profinite Groups

Abstract

We prove some conditions on a given abstract group G, such that the group Aut_c(\(\hat G\)) of the continuous automorphisms of the profinite completion \(\hat G\) of G endowed with the congruence subgroup topology, is profinite. Also, for a given abstract group G, if Aut_c(\(\hat G\)) is profinite, then we establish relations betweenG, Aut(G), \(\widehat {Aut(G)}\), and Aut_c(\(\hat G\)) when each of these groups is endowed with appropriate topology. Finally, we applied the obtained results to the class of one-relator groups given by the presentation G_mn = 〈a, b; [a^m, bⁿ] = 1〉 (m > 1,n > 1).