Locally cyclic group
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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.
- 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.
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.
- Let p be any prime, and let μp∞ denote the set of all pth-power roots of unity in C, i.e.
- <math>\mu_{p^{\infty}} = \left\{ \exp\left(\frac{2\pi im}{p^{k}}\right) : m,k\in\mathbb{Z}\right\}<math>
- Then μp∞ is locally cyclic but not cyclic.
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.