Finitely presented implies all homomorphisms to any finite group can be listed in finite time

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely presented group) must also satisfy the second group property (i.e., finitely generated group for which all homomorphisms to any finite group can be listed in finite time)
View all group property implications | View all group property non-implications
Get more facts about finitely presented group|Get more facts about finitely generated group for which all homomorphisms to any finite group can be listed in finite time

Statement

Suppose $G$ is a finitely presented group having a finite presentation using a generating set $S$ and relation set $R$. Then, for any finite group $K$ and any set map from $S$ to $K$, it is possible to determine in finite time whether the set map extends to a homomorphism from $G$ to $K$.

Note that this implies that all homomorphisms to $K$ can be listed in finite time, because there are only finitely many candidate set maps from $S$ to $K$.

Related facts

Similar facts

• Finitely generated implies finitely many homomorphisms to any finite group

Applications

• Finitely presented and residually finite implies solvable word problem
• Finitely presented and conjugacy-separable implies solvable conjugacy problem

Proof

Proof idea

The idea is as follows: for any relation word (i.e., every element of $R$) check whether the image of that word, viewed as an element of $K$, is the identity element of $K$.