Geodesic
The normalizer condition
Matematičeskie zametki, Tome 3 (1968) no. 1, pp. 45-50.

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It is proved that a torsion-free group $\mathfrak G$, in which every infinite proper subgroup is distinct from its normalizer satisfies the normalizer condition, i.e., every proper subgroup is distinct from its normalizer. 
@article{MZM_1968_3_1_a5,
     author = {S. N. Chernikov},
     title = {The normalizer condition},
     journal = {Matemati\v{c}eskie zametki},
     pages = {45--50},
     publisher = {mathdoc},
     volume = {3},
     number = {1},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1968_3_1_a5/}
}
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S. N. Chernikov. The normalizer condition. Matematičeskie zametki, Tome 3 (1968) no. 1, pp. 45-50. http://geodesic.mathdoc.fr/item/MZM_1968_3_1_a5/