In mathematics, a locally cyclic group is a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

(''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 In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

and

quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...

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 In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial ordering being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, ...

is distributive . * The torsion-free rank of a locally cyclic group is 0 or 1. * The

endomorphism ring In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...

of a locally cyclic group is

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

Examples of locally cyclic groups that are not cyclic

Examples of abelian groups that are not locally cyclic

* The additive group of

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s (R, +); the subgroup generated by 1 and (comprising all numbers of the form ''a'' + ''b'') is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

to the

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...

Z + Z, which is not cyclic.

References

*. *. * {{Refend Abelian group theory Properties of groups

Some facts

Examples of locally cyclic groups that are not cyclic

Examples of abelian groups that are not locally cyclic

See also

References