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Locally cyclic group
From Wikipedia, the free encyclopedia
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In mathematics, a locally cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic.
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Some facts
- Every cyclic group is locally cyclic, and every locally cyclic group is abelian.[1]
- 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.[citation needed]
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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.
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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.
See also
References
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