Locally cyclic group

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

Some facts

• Every cyclic group is locally cyclic, and every locally cyclic group is abelian.
• Every finitely-generated locally cyclic group is cyclic.
• Every subgroup and quotient group of a locally cyclic group is locally cyclic.
• Every homomorphic image of a locally cyclic group is locally cyclic.
• A group is locally cyclic if and only if every pair of elements in the group generates a cyclic group.
• A group is locally cyclic if and only if its lattice of subgroups is distributive (Ore 1938).
• The torsion-free rank of a locally cyclic group is 0 or 1.
• The endomorphism ring of a locally cyclic group is commutative.

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/2^b, is also locally cyclic – any pair of dyadic rational numbers a/2^b and c/2^d is contained in the cyclic subgroup generated by 1/2^max(b,d).
• Let p be any prime, and let μ_p^∞ denote the set of all pth-power roots of unity in C, i.e.

  [math]\displaystyle{ \mu_{p^\infty} = \left\{\exp\left(\frac{2\pi im}{p^k}\right) : m, k \in \mathbb{Z}\right\} }[/math]

  Then μ_p^∞ 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, +); the subgroup generated by 1 and π (comprising all numbers of the form a + bπ) is isomorphic to the direct sum Z + Z, which is not cyclic.

References

• "19.2 Locally Cyclic Groups and Distributive Lattices", Theory of Groups, American Mathematical Society, 1999, pp. 340–341, ISBN 978-0-8218-1967-8 .

• "Structures and group theory. II", Duke Mathematical Journal 4 (2): 247–269, 1938, doi:10.1215/S0012-7094-38-00419-3, http://dml.cz/bitstream/handle/10338.dmlcz/100155/CzechMathJ_05-1955-3_8.pdf .

• Rose, John S. (2012). A Course on Group Theory. Dover Publications. ISBN 978-0-486-68194-8.