• Source: 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






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

Kata Kunci Pencarian:

• Grup siklik lokal
• Locally cyclic group
• Cyclic group
• Glossary of group theory
• Finitely generated group
• Locally compact abelian group
• Subgroups of cyclic groups
• Profinite group
• List of group theory topics
• Cyclic order
• List of abstract algebra topics