# Group in which every finite subgroup is cyclic

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

## 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 $G$ is a group in which every finite subgroup is cyclic, and $H$ is a subgroup of $G$, then $H$ 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 $G$ in which every finite subgroup is cyclic, and a normal subgroup $H$ of $G$ such that the quotient group $G/H$ has a finite subgroup that is not cyclic. For instance, $G = \mathbb{Z} \times \mathbb{Z}$ and $H = 2\mathbb{Z} \times 2\mathbb{Z}$
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 $G_2$ have the property that every finite subgroup is cyclic, but $G_1 \times G_2$ does not. For instance, set both $G_1$ and $G_2$ 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 $n$, there are at most $n$ elements of order dividing $n$ |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