# Group satisfying Tits alternative

From Groupprops

## Contents

## Definition

A group is said to satisfy the **Tits alternative** if for every 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).

## Relation with other properties

### Stronger properties

- Finite group
- Abelian group
- Nilpotent group
- Solvable group
- Virtually Abelian group
- Virtually nilpotent group
- Virtually solvable group
- Free group