Group having no free non-abelian subgroup

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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Definition

A group having no free non-abelian subgroup is a group satisfying the following equivalent conditions:

• It has no subgroup that is a free non-abelian subgroup.
• It has no subgroup isomorphic to the free group on two generators.
• For fixed $r>1$, it has no subgroup isomorphic to a free group on $r$ generators.
• It has no subgroup isomorphic to a free group on a countably infinite number of generators.

Relation with other properties

Stronger properties

• Finite group
• Abelian group
• Nilpotent group
• Virtually abelian group
• Virtually nilpotent group
• Virtually solvable group
• Periodic group
• Group satisfying a nontrivial identity