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

From Groupprops

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)

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## Statement

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

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

## Related facts

### Similar facts

### 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 ) check whether the image of that word, viewed as an element of , is the identity element of .