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Locally cyclic group

1,275 bytes added, 22:29, 11 April 2017
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===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 <math>G</math> is a locally cyclic group and <math>H</math> is a subgroup of <math>G</math>, then <math>H</math> is also a locally cyclic group.
| [[satisfies metaproperty::quotient-closed group property]] || Yes || || If <math>G</math> is a locally cyclic group and <math>H</math> is a normal subgroup of <math>G</math>, then the quotient group <math>G/H</math> is also a locally cyclic group.
| [[dissatisfies metaproperty::finite direct product-closed group property]] || No || See next column || It is possible to have groups <math>G_1, G_2</math> such that both <math>G_1</math> and <math>G_2</math> are locally cyclic but the [[external direct product]] <math>G_1 \times G_2</math> is not locally cyclic. In fact, ''any'' choice of nontrivial <math>G_1, G_2</math> gives an example.
==Relation with other properties==
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