Locally cyclic group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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

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
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

Weaker properties conditional to nontriviality

The following properties are weaker if we restrict to nontrivial groups.

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
splitting-simple group has no proper nontrivial complemented normal subgroup |FULL LIST, MORE INFO
directly indecomposable group has no proper nontrivial direct factor |FULL LIST, MORE INFO