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 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 |