Examples of Locally Cyclic Groups That Are Not Cyclic
- The additive group of rational numbers (Q, +) is locally cyclic – any pair of rational numbers a/b and c/d is contained in the cyclic subgroup generated by 1/bd.
- The additive group of the dyadic rational numbers, the rational numbers of the form a/2b, is also locally cyclic – any pair of dyadic rational numbers a/2b and c/2d is contained in the cyclic subgroup generated by 1/2max(b,d).
- Let p be any prime, and let μp∞ denote the set of all pth-power roots of unity in C, i.e.
- Then μp∞ is locally cyclic but not cyclic. This is the Prüfer p-group. The Prüfer 2-group is closely related to the dyadic rationals (it can be viewed as the dyadic rationals modulo 1).
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