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New page: {{group property}} ==Definition== A group is termed '''locally cyclic''' if it satisfies the following equivalent conditions: # Every finitely generated...

{{group property}}

==Definition==

A [[group]] is termed '''locally cyclic''' if it satisfies the following equivalent conditions:

# Every [[finitely generated group|finitely generated]] subgroup of the group is [[cyclic group|cyclic]].

# The subgroup generated by any two elements of the group is cyclic.

# Its [[lattice of subgroups]] is a distributive lattice. In other words, the operations of [[join of subgroups]] and [[intersection of subgroups]] distribute over each other.

==Relation with other properties==

===Stronger properties===

* [[Weaker than::Cyclic group]]

===Weaker properties===

* [[Stronger than::Abelian group]]

* [[Stronger than::Group in which every finite subgroup is cyclic]]

==Definition==

A [[group]] is termed '''locally cyclic''' if it satisfies the following equivalent conditions:

# Every [[finitely generated group|finitely generated]] subgroup of the group is [[cyclic group|cyclic]].

# The subgroup generated by any two elements of the group is cyclic.

# Its [[lattice of subgroups]] is a distributive lattice. In other words, the operations of [[join of subgroups]] and [[intersection of subgroups]] distribute over each other.

==Relation with other properties==

===Stronger properties===

* [[Weaker than::Cyclic group]]

===Weaker properties===

* [[Stronger than::Abelian group]]

* [[Stronger than::Group in which every finite subgroup is cyclic]]

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