# Group in which every finite abelian 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 abelian subgroup is cyclic is a group satisfying the following equivalent conditions:**

- Every Finite abelian subgroup (?) (i.e., every subgroup that is a Finite abelian group (?)) is cyclic as a group: in other words, is a Cyclic subgroup (?).
- Every Finite subgroup (?) is a Finite group with periodic cohomology (?).

## 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 abelian subgroup) satisfies the second property (finite cyclic subgroup), and vice versa.

View other group properties obtained in this way

## Relation with other properties

### Stronger properties

- Finite group with periodic cohomology: A finite group with periodic cohomology is characterized by every abelian subgroup being cyclic.
- Group in which every finite subgroup is cyclic
- Group with at most n elements of order dividing n
- Group with at most n pairwise commuting elements of order dividing n
- Multiplicative group of a division ring