Group in which every finite abelian subgroup is cyclic

Definition
A '''group in which every finite abelian subgroup is cyclic is a group satisfying the following equivalent conditions:


 * Every fact about::finite abelian subgroup (i.e., every subgroup that is a fact about::finite abelian group) is cyclic as a group: in other words, is a fact about::cyclic subgroup.
 * Every fact about::finite subgroup is a fact about::finite group with periodic cohomology.

Stronger properties

 * Weaker than::Finite group with periodic cohomology: A finite group with periodic cohomology is characterized by every abelian subgroup being cyclic.
 * Weaker than::Group in which every finite subgroup is cyclic
 * Weaker than::Group with at most n elements of order dividing n
 * Weaker than::Group with at most n pairwise commuting elements of order dividing n
 * Weaker than::Multiplicative group of a skew field