# Changes

## Locally cyclic group

, 22:29, 11 April 2017
no edit summary
==Definition==

===Symbol-free definition===
A [[group]] is termed '''locally cyclic''' if it satisfies the following equivalent conditions:
(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]].

== Examples ==

* For [[finite group]]s, this coincide with being a [[cyclic group]] (and hence, a [[finite cyclic group]]).
* For infinite groups, this could be the [[group of integers]] (the only infinite cyclic group), a [[quasicyclic group]], or any other subquotient of the [[group of rationals]].

== Metaproperties ==

{| class="sortable" border="1"
! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
|-
| [[satisfies metaproperty::subgroup-closed group property]] || Yes || || If $G$ is a locally cyclic group and $H$ is a subgroup of $G$, then $H$ is also a locally cyclic group.
|-
| [[satisfies metaproperty::quotient-closed group property]] || Yes || || If $G$ is a locally cyclic group and $H$ is a normal subgroup of $G$, then the quotient group $G/H$ is also a locally cyclic group.
|-
| [[dissatisfies metaproperty::finite direct product-closed group property]] || No || See next column || It is possible to have groups $G_1, G_2$ such that both $G_1$ and $G_2$ are locally cyclic but the [[external direct product]] $G_1 \times G_2$ is not locally cyclic. In fact, ''any'' choice of nontrivial $G_1, G_2$ gives an example.
|}
==Relation with other properties==