On groups all of whose proper subgroups of which are finite cyclic

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.