On infinite $p$-groups with cyclic subgroups

For each odd prime $p$, continuum many nonisomorphic simple groups are constructed having isomorphic subgroup lattices and having the property that every proper subgroup is a cyclic $p$-group. Also constructed is a periodic group of infinite width where every proper subgroup is cyclic. The proofs are based on papers by A. Yu. Ol'shanskii.
Figures: 2.
Bibliography: 6 titles.