Difference between revisions of "Locally cyclic group"

From Groupprops
Jump to: navigation, search
(New page: {{group property}} ==Definition== A group is termed '''locally cyclic''' if it satisfies the following equivalent conditions: # Every finitely generated...)
 
Line 2: Line 2:
  
 
==Definition==
 
==Definition==
 +
 +
===Symbol-free definition===
  
 
A [[group]] is termed '''locally cyclic''' if it satisfies the following equivalent conditions:
 
A [[group]] is termed '''locally cyclic''' if it satisfies the following equivalent conditions:
Line 7: Line 9:
 
# Every [[finitely generated group|finitely generated]] subgroup of the group is [[cyclic group|cyclic]].
 
# 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.
 
# 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.
 
# 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==
 
==Relation with other properties==

Revision as of 13:13, 6 January 2009

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

Symbol-free definition

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

  1. Every finitely generated subgroup of the group is cyclic.
  2. The subgroup generated by any two elements of the group is cyclic.
  3. It is isomorphic to a subquotient (i.e., a quotient group of a subgroup) of the group of rational numbers.
  4. 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 information: 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

Stronger properties

Weaker properties