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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.

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