Some facts

• Every cyclic group is locally cyclic, and every locally cyclic group is abelian
  Abelian group
  In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
  
  .
• Every finitely-generated locally cyclic group is cyclic.
• Every subgroup
  Subgroup
  In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
  
   and quotient group
  Quotient group
  In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
  
   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
  Lattice of subgroups
  In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion....
  
   is distributive
  Distributive lattice
  In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
  
   .
• The torsion-free rank of a locally cyclic group is 0 or 1.

Examples of locally cyclic groups that are not cyclic

• The additive group of rational number
  Rational number
  In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
  
  s (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/2^b, is also locally cyclic – any pair of dyadic rational numbers a/2^b and c/2^d is contained in the cyclic subgroup generated by 1/2^max(b,d).
• Let p be any prime, and let μ_p^∞ denote the set of all pth-power roots of unity
  Root of unity
  In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
  
   in C, i.e.

Then μ_p^∞ is locally cyclic but not cyclic. This is the Prüfer p-group

Prüfer group

In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z, for a prime number p is the unique p-group in which every element has p pth roots. The group is named after Heinz Prüfer...

. 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

• The additive group of real number
  Real number
  In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
  
  s (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
  Group isomorphism
  In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
  
   to the direct sum Z + Z, and this group is not cyclic.