Geodesic
On groups all of whose proper subgroups of which are finite cyclic
Izvestiya. Mathematics , Tome 39 (1992) no. 2, pp. 905-957

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For any odd number $n\geqslant 1003$, the authors construct an infinite 2-generator group each of whose proper subgroups is contained in a cyclic subgroup of order $n$. This result strengthens analogous results of Ol'shanskii for prime $n>10^{75}$ and Atabekyan and Ivanov for odd $n>10^{80}$. The proof is carried out in the original language of Novikov–Adyan theory. 
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     author = {S. I. Adian and I. G. Lysenok},
     title = {On groups all of whose proper subgroups of which are finite cyclic},
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S. I. Adian; I. G. Lysenok. On groups all of whose proper subgroups of which are finite cyclic. Izvestiya. Mathematics , Tome 39 (1992) no. 2, pp. 905-957. http://geodesic.mathdoc.fr/item/IM2_1992_39_2_a0/