Locally cyclic torsion-free group

From Groupprops
Jump to: navigation, search
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: locally cyclic group and torsion-free group
View other group property conjunctions OR view all group properties

Definition

A group is termed a locally cyclic torsion-free group if it satisfies the following equivalent conditions:

  1. It is both a locally cyclic group (i.e., every finitely generated subgroup is cyclic) and a torsion-free group (i.e., no non-identity element has finite order).
  2. It is isomorphic to a subgroup of the group of rational numbers.

Equivalence of definitions

Further information: Equivalence of definitions of locally cyclic torsion-free group

Relation with other properties

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Locally cyclic group every finitely generated subgroup is cyclic |FULL LIST, MORE INFO