# Group in which every finite subgroup is cyclic

From Groupprops

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 propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A **group in which every finite subgroup is cyclic** is a group where every finite subgroup (i.e., subgroup that is a finite group) is cyclic.

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | Yes | If is a group in which every finite subgroup is cyclic, and is a subgroup of , then is also a group in which every finite subgroup is cyclic. | |

quotient-closed group property | No | See next column | It is possible to have a group in which every finite subgroup is cyclic, and a normal subgroup of such that the quotient group has a finite subgroup that is not cyclic. For instance, and |

finite direct product-closed group property | No | See next column | It is possible to have groups such that both and have the property that every finite subgroup is cyclic, but does not. For instance, set both and as equal to cyclic group:Z2. |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

locally cyclic group | every finitely generated subgroup is cyclic | Group with at most n elements of order dividing n|FULL LIST, MORE INFO | ||

multiplicative group of a field | occurs as the multiplicative group of a field | |FULL LIST, MORE INFO | ||

group with at most n elements of order dividing n | for any positive integer , there are at most elements of order dividing | |FULL LIST, MORE INFO | ||

torsion-free group | no non-identity element of finite order | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

group in which every finite abelian subgroup is cyclic | every finite abelian subgroup is cyclic | |FULL LIST, MORE INFO |

## Formalisms

### In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (finite subgroup) satisfies the second property (finite cyclic subgroup), and vice versa.

View other group properties obtained in this way