Locally cyclic group

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 .
• 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/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 μ**p∞ denote the set of all pth-power roots of unity in C, i.e. : \mu_{p^\infty} = \left{\exp\left(\frac{2\pi im}{p^k}\right) : m, k \in \mathbb{Z}\right}

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 (comprising all numbers of the form a + b) is isomorphic to the direct sum Z + Z, which is not cyclic.

References

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