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Locally cyclic group

539 bytes added, 13:13, 6 January 2009
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==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==
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