# Group having no free non-abelian subgroup

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

## 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 , it has no subgroup isomorphic to a free group on generators.
- It has no subgroup isomorphic to a free group on a countably infinite number of generators.