Geodesic
Аn example of the $G$-periodic torsion free group
Algebra i logika, Tome 6 (1967) no. 3, pp. 5-7.

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A group $G$ is called $G$-periodic if for any element $g$ of $G$ there exist the elements $h_1, h_2, \dots, h_k$ such that $$ (h_1^{-1}gh_1)(h_2^{-1}gh_2)\dots(h_k^{-1}gh_k)=1. $$ In this note an example of $G$-periodic torsion free group is constructed. 
@article{AL_1967_6_3_a0,
     author = {Yu. M. Gor\v{c}akov},
     title = {{\CYRA}n example of the $G$-periodic torsion free group},
     journal = {Algebra i logika},
     pages = {5--7},
     publisher = {mathdoc},
     volume = {6},
     number = {3},
     year = {1967},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_1967_6_3_a0/}
}
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Yu. M. Gorčakov. Аn example of the $G$-periodic torsion free group. Algebra i logika, Tome 6 (1967) no. 3, pp. 5-7. http://geodesic.mathdoc.fr/item/AL_1967_6_3_a0/