On periodic groups of odd period n ≥ 1003

In the paper, using the Adyan-Lysenok theorem claiming that, for any odd nurnber n > 1003, there is an infinite group each of whose proper Subgroups is contained in a cyclic subgroup of order 71, 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 >= k >= 2 and for any odd n >= 1003, 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.