Difference between revisions of "Locally cyclic group"

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(New page: {{group property}} ==Definition== A group is termed '''locally cyclic''' if it satisfies the following equivalent conditions: # Every finitely generated...)
 
(Weaker properties)
 
(10 intermediate revisions by the same user not shown)
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# 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.
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# 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.
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===Equivalence of definitions===
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{{further|[[Locally cyclic iff subquotient of rationals]], [[Locally cyclic iff distributive lattice of subgroups]]}}
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(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]].
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== Examples ==
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* For [[finite group]]s, this coincide with being a [[cyclic group]] (and hence, a [[finite cyclic group]]).
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* 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]].
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== Metaproperties ==
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{| class="sortable" border="1"
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! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
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|-
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| [[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.
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|-
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| [[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.
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|-
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| [[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.
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|-
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| [[satisfies metaproperty::lattice-determined group property]] || Yes || [[locally cyclic iff distributive lattice of subgroups]] || Given two [[lattice-isomorphic groups]] <math>G_1, G_2</math>, either both <math>G_1</math> and <math>G_2</math> are locally cyclic or neither is. The explicit condition for being locally cyclic is that the lattice of subgroups is distributive.
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|}
  
 
==Relation with other properties==
 
==Relation with other properties==
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===Stronger properties===
 
===Stronger properties===
  
* [[Weaker than::Cyclic group]]
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{| class="sortable" border="1"
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
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|-
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| [[Weaker than::cyclic group]] || generated by one element || || [[locally cyclic not implies cyclic]] || {{intermediate notions short|locally cyclic group|cyclic group}}
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|}
  
 
===Weaker properties===
 
===Weaker properties===
  
* [[Stronger than::Abelian group]]
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{| class="sortable" border="1"
* [[Stronger than::Group in which every finite subgroup is cyclic]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
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|-
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| [[Stronger than::abelian group]] || any two elements commute || [[locally cyclic implies abelian]] || [[abelian not implies locally cyclic]] || {{intermediate notions short|abelian group|locally cyclic group}}
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|-
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| [[Stronger than::epabelian group]] || abelian and the Schur multiplier is trivial || [[locally cyclic implies epabelian]] || || {{intermediate notions short|epabelian group|locally cyclic group}}
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|-
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| [[Stronger than::group whose automorphism group is abelian]] || automorphism group is abelian || [[locally cyclic implies abelian automorphism group]] || [[abelian and abelian automorphism group not implies locally cyclic]] || {{intermediate notions short|aut-abelian group|locally cyclic group}}
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|-
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| [[Stronger than::group with at most n elements of order dividing n]] || || || || {{intermediate notions short|group with at most n elements of order dividing n|locally cyclic group}}
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|-
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| [[Stronger than::group in which every finite subgroup is cyclic]] || every finite subgroup is cyclic || || || {{intermediate notions short|group in which every finite subgroup is cyclic|locally cyclic group}}
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|}

Latest revision as of 22:55, 1 July 2017

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

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.

Examples

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
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.
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.
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.
lattice-determined group property Yes locally cyclic iff distributive lattice of subgroups Given two lattice-isomorphic groups G_1, G_2, either both G_1 and G_2 are locally cyclic or neither is. The explicit condition for being locally cyclic is that the lattice of subgroups is distributive.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
cyclic group generated by one element locally cyclic not implies cyclic |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
abelian group any two elements commute locally cyclic implies abelian abelian not implies locally cyclic Epabelian group|FULL LIST, MORE INFO
epabelian group abelian and the Schur multiplier is trivial locally cyclic implies epabelian |FULL LIST, MORE INFO
group whose automorphism group is abelian automorphism group is abelian locally cyclic implies abelian automorphism group abelian and abelian automorphism group not implies locally cyclic |FULL LIST, MORE INFO
group with at most n elements of order dividing n |FULL LIST, MORE INFO
group in which every finite subgroup is cyclic every finite subgroup is cyclic Group with at most n elements of order dividing n|FULL LIST, MORE INFO