Locally cyclic group

In mathematics, a locally cyclic group is a group (G, *) in which every finitely generated subgroup is cyclic.

Table of contents

1 Some facts 2 Examples of locally cyclic groups that are not cyclic 3 Examples of abelian groups that are not locally 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 μ_p^∞ denote the set of all pth-power roots of unity in C, i.e.

Then μ_p^∞ 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.