# Group satisfying Tits alternative for finitely generated subgroups

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 is said to satisfy the **Tits alternative for finitely generated subgroups** if for every finitely generated subgroup of it, one of these two conditions holds:

- The subgroup is virtually solvable (i.e., has a solvable subgroup of finite index)
- The subgroup contains a free non-abelian subgroup (which is equivalent to saying that it contains a copy of free group:F2).