Locally cyclic group

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

Some facts[edit]

  • 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.

Examples of locally cyclic groups that are not cyclic[edit]

  • 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.
  • 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.
\mu_{p^{\infty}} = \left\{ \exp\left(\frac{2\pi im}{p^{k}}\right) : m,k\in\mathbb{Z}\right\}
Then μp is locally cyclic but not cyclic. This is the Prüfer p-group. The Prüfer 2-group is closely related to the dyadic rationals (it can be viewed as the dyadic rationals modulo 1).

Examples of abelian groups that are not locally cyclic[edit]

  • 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.

References[edit]