Bureaucrats, emailconfirmed, Administrators

38,756

edits
no edit summary

==Definition==

===Symbol-free 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.

# It is isomorphic to a [[subquotient]] (i.e., a [[quotient group]] of a [[subgroup]]) of the [[group of rational numbers]].

# 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.

===Equivalence of definitions===

{{further|[[Locally cyclic iff subquotient of rationals]], [[Locally cyclic iff distributive lattice of subgroups]]}}

(1) and (2) are clearly equivalent. For the equivalence of (1) and (2) with (3), refer [[locally cyclic iff subquotient of rationals]]. For the equivalence with (4), refer [[locally cyclic iff distributive lattice of subgroups]].

==Relation with other properties==

Retrieved from "https://groupprops.subwiki.org/wiki/Special:MobileDiff/15495"