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On periodic groups of odd period $$n\geq 1003$$. (English. Russian original) Zbl 1152.20034
Math. Notes 82, No. 4, 443-447 (2007); translation from Mat. Zametki 82, No. 4, 495-500 (2007).
Summary: Using the Adyan-Lysenok theorem claiming that, for any odd number $$n\geq 1003$$, 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 $$m\geq k\geq 2$$ and for any odd $$n\geq 1003$$, 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.

##### MSC:
 20F50 Periodic groups; locally finite groups 20F05 Generators, relations, and presentations of groups 20E28 Maximal subgroups 20E07 Subgroup theorems; subgroup growth 20E32 Simple groups
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