Definition
A finitely generated group for which all homomorphisms to any finite group can be listed in finite time is a finitely generated group such that the following holds for one (and hence every) finite generating set of :
 For any finite group explicitly specified by means of its multiplication table, and every set map from to , it is possible to determine in finite time whether the est map extends to a group homomorphism from to . Note that if it does extend, it extends uniquely.
 For any finite group explicitly specified by means of its multiplication table, it is possible to, in finite time, list all the set maps from to that extend to group homomorphisms from to .
 For any finite group explicitly specified by means of its multiplication table, and every set map from to , it is possible to determine in finite time whether the est map extends to a surjective group homomorphism from to . Note that if it does extend, it extends uniquely.
 For any finite group explicitly specified by means of its multiplication table, it is possible to, in finite time, list all the set maps from to that extend to surjective group homomorphisms from to .
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 properties
VIEW RELATED: Group property implications  Group property nonimplications Group metaproperty satisfactions  Group metaproperty dissatisfactions  Group property satisfactions  Group property dissatisfactions
Relation with other properties
Stronger properties
Weaker properties
Property 
Meaning 
Proof of implication 
Proof of strictness (reverse implication failure) 
Intermediate notions


finitely generated group 



