Classes of topological groups suggested by Galois theory (Q1174824)

Let \(G\) be a profinite abelian group. Motivated by the Galois theory of infinite extensions, cf. \textit{H. Bass} and the author [J. Indian Math. Soc., New Ser. 36, 1-7 (1972; Zbl 0284.20037)], one is interested in dense totally bounded subgroups \(H\) of \(G\) such that all subgroups of \(H\) are closed in \(H\). The author proves that \(G\) and \(H\) satisfy these conditions if and only if \(G\) is the profinite completion of \(H\), and \(H\) is an extension of a free abelian group of finite rank by a torsion group with primary parts of finite exponents. The second part of the paper contains various general results on topologically complete groups, i.e. topological groups with trivial center such that every topological automorphism is an inner automorphism. The structure of profinite abelian groups which admit only one compact group topology has been determined by \textit{A. Hulanicki} [Diss. Math. (Rozprawy Mat.) 38, 1--58 (1964; Zbl 0119.03301)]. In theorem 3.1, the author describes all locally compact group topologies on these groups. Finally, he extends this description to (possibly nonabelian) profinite groups with Sylow subgroups of similar shape.