Locally cyclic group

In group theory, a locally cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic.

Some facts

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.[2]
  • 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).

Examples of abelian groups that are not locally cyclic

  • The additive group of real numbers (R, +) is not locally cyclic—the subgroup generated by 1 and π consists of all numbers of the form a + bπ. This group is isomorphic to the direct sum Z + Z, and this group is not cyclic.

References

  1. Rose (2012), p. 54.
  2. Rose (2012), p. 52.
  • Hall, Marshall, Jr. (1999), "19.2 Locally Cyclic Groups and Distributive Lattices", Theory of Groups, American Mathematical Society, pp. 340–341, ISBN 978-0-8218-1967-8.
  • Rose, John S. (2012) [unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978]. A Course on Group Theory. Dover Publications. ISBN 0-486-68194-7.
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