# Finitely generated group for which all homomorphisms to any finite group can be listed in finite time

From Groupprops

## Contents

## 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 propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finitely presented group | |FULL LIST, MORE INFO | |||

finitely generated simple group | |FULL LIST, MORE INFO | |||

finite group | |FULL LIST, MORE INFO | |||

finitely generated free group | |FULL LIST, MORE INFO | |||

finitely generated abelian group | |FULL LIST, MORE INFO | |||

finitely generated nilpotent group | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finitely generated group |