# Locally cyclic group

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

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

(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.

## 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