Locally cyclic group

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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

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

• For finite groups, 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

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