# SQ-universal group

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

## Contents

## Definition

### Symbol-free definition

A group is termed **SQ-universal** with respect to finitely generated groups (or simple **SQ-universal**) if it satisfies the following equivalent conditions:

- Every finitely generated group is a subquotient of the given group
- There exists a finitely generated free group (on at least two generators) that occurs as a subquotient of the given group
- The free group on two generators is a subquotient of the given group

## Relation with other properties

### Stronger properties

- Non-Abelian finitely generated free group (i.e. a finitely generated free group on at least two generators)