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

From Groupprops
Jump to: navigation, search

Definition

A finitely generated group for which all homomorphisms to any finite group can be listed in finite time is a finitely generated group G such that the following holds for one (and hence every) finite generating set S of G:

  1. For any finite group K explicitly specified by means of its multiplication table, and every set map from S to K, it is possible to determine in finite time whether the est map extends to a group homomorphism from G to K. Note that if it does extend, it extends uniquely.
  2. For any finite group K explicitly specified by means of its multiplication table, it is possible to, in finite time, list all the set maps from S to K that extend to group homomorphisms from G to K.
  3. For any finite group K explicitly specified by means of its multiplication table, and every set map from S to K, it is possible to determine in finite time whether the est map extends to a surjective group homomorphism from G to K. Note that if it does extend, it extends uniquely.
  4. For any finite group K explicitly specified by means of its multiplication table, it is possible to, in finite time, list all the set maps from S to K that extend to surjective group homomorphisms from G to K.
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 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
Retrieved from "https://groupprops.subwiki.org/w/index.php?title=Finitely_generated_group_for_which_all_homomorphisms_to_any_finite_group_can_be_listed_in_finite_time&oldid=38525"