0. INTRODUCTION A topology T on a group G is a group topology if both group operations of G, (dh) i-- g - h and g \-t g~x, are T-continuous, i.e. if (G,T) is a topological group. In what follows all topological groups and group topologies considered are assumed to be Hausdorff. Clearly the discrete topology on a group is a group topology. In 1945 Markov [63] asked whether every infinite group admits a non-discrete group topology. In 1953 Kertesz and Szele [57] gave a positive answer to this question for Abelian groups. In the general case this question, became known later as Markov's problem, re- mained unsolved until 1980 when Shelah [78] applied deep results from the theory of groups with small cancellation of words to give an uncountable counterexample to Markov's problem assuming the continuum hypothesis (the recourse to the con- tinuum hypothesis was avoided later by Hesse [48]). In somewhat the same time Ol'shanskii [69] observed that appropriate quotients of the famous Adian's groups A(m, n) can serve as an example of a countable infinite group possessing no group topology beyond the discrete one, without any additional set-theoretic assumptions. In 1944 Halmos [44] asked for a characterization of the Abelian groups admitting a compact group topology. This problem was partially solved by I. Kaplansky [55] and completely by Harrison [45] and Hulanicki [52]. For future references we remind briefly their results (see [50, §25]): 0.1 Theorem, (i) Let G be an Abelian group and G = D(G)®R(G) (1) be its canonical decomposition into the divisible part D(G) and the reduced part R(G) = G/D(G). (A group is reduced if it does not contain a non-trivial divisible 1
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