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I found a result in Cardinal invariants and independence results in the poset of precompact group topologies, Alessandro Berarducci, Dikran Dikranjan, Marco Forti, Stephen Watson: Let $G$ be an infinite abelian group. The power set of the power set of $G$ can be embedded in the poset of all Hausdorff group topologies on $G$. So there are $A,B\in ... 1 On random generations there are many results and you are probably familiar with them. Regrading your question, you should look at Belazs Szegedy's paper on the Nottingham group, http://blms.oxfordjournals.org/content/37/1/75. There are also results of Tsachik Gelander, Jonathan Barlev and Emanuel Breuillard on such questions. 0 I found this proposition: On any Abelian group of cardinality$α$, one can induce$2^{2^\alpha}$-many distinct totally bounded group topologies. in S. Berhanu, W.W. Comfort, J.D. Reid, Counting subgroups and topological group topologies, Pacific J. Math. 116 (2) (1985) 217–241, MR 86f:20069. There are at most$\mathfrak c\$ sequences of subgroups on ...