On Periodic Groups of Odd Period $n\ge1003$

In the paper, using the Adyan–Lysenok theorem claiming that, for any odd number $n\ge1003$, there is an infinite group each of whose proper subgroups is contained in a cyclic subgroup of order $n$, it is proved that the set of groups with this property has the cardinality of the continuum (for a given $n$). Further, it is proved that, for $m\ge k\ge2$ and for any odd $n\ge1003$, the $m$-generated free $n$-periodic group is residually both a group of the above type and a $k$-generated free $n$-periodic group, and it does not satisfy the ascending and descending chain conditions for normal subgroups either.