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