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On the number of group topologies on countable groups
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2014), pp. 101-112 Cet article a éte moissonné depuis la source Math-Net.Ru

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If a countable group $G$ admits a non-discrete Hausdorff group topology, then the lattice of all group topologies of the group $G$ admits: – continuum $c$ of non-discrete metrizable group topologies such that $\sup\{\tau_1,\tau_2\}$ is the discrete topology for any two of these topologies; – two to the power of continuum of coatoms in the lattice of all group topologies. 
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V. I. Arnautov; G. N. Ermakova. On the number of group topologies on countable groups. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2014), pp. 101-112. http://geodesic.mathdoc.fr/item/BASM_2014_1_a6/

[1] Arnautov V. I., Ermakova G. N., “On the number of metrizable group topologies on countable groups”, Bul. Acad. Ştiinţe Repub. Moldova, Matematica, 2013, no. 2(72)–3(73), 17–26

[2] Bourbaki N., Topologie generale, Moskva, 1958 (in Russian)

[3] Engelking R., General topology, Moskva, 1986 (in Russian) | MR