# Finitely presented and residually finite implies solvable word problem

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 residually finite group) must also satisfy the second group property (i.e., group with solvable word problem)

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## Statement

Any finitely presented residually finite group (i.e., a group that is both a finitely presented group and a residually finite group) is a group with solvable word problem.

## Related facts

### Similar facts

## Facts used

- Finitely presented implies all homomorphisms to any finite group can be listed in finite time
- Residually finite and all homomorphisms to any finite group can be listed in finite time implies solvable word problem

## Proof

The proof follows directly by combining Facts (1) and (2).