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
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VIEW 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)