Group in which every finite subgroup is cyclic

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			|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