This has some interesting consequences. For instance, the Higman embedding theorem can be used to construct a group containing an isomorphic copy of every finitely presented group with solvable word problem. It seems natural to ask whether this group can have solvable word problem. But it is a consequence of the Boone-Rogers result that: Corollary: There is no universal solvable word problem group. That is, if G is a finitely presented group that contains an isomorphic copy of every finitely presented group with solvable word problem, then G itself must have unsolvable word problem. Remark: Suppose G = ⟨X|R⟩ is a finitely presented group with solvable word problem and H is a finite subset of G. Let H* = ⟨H⟩, be the group generated by H. Then the word problem in H* is solvable: given two words h, k in the generators H of H*, write them as words in X and compare them using the solution to the word problem in G. It is easy to think that this demonstrates a uniform solution of the word problem for the class K (say) of finitely generated groups that can be embedded in G. If this were the case, the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers. However, the solution just exhibited for the word problem for groups in K is not uniform. To see this, consider a group J = ⟨Y|T⟩ ∈ K; in order to use the above argument to solve the word problem in J, it is first necessary to exhibit a mapping e: Y → G that extends to an embedding e*: J → G. If there were a recursive function that mapped (finitely generated) presentations of groups in K to embeddings into G, then a uniform solution of the word problem in K could indeed be constructed. But there is no reason, in general, to suppose that such a recursive function exists. However, it turns out that, using a more sophisticated argument, the word problem in J can be solved without using an embedding e: J → G. Instead an enumeration of homomorphisms is used, and since such an enumeration can be constructed uniformly, it results in a uniform solution to the word problem in K. Proof that there is no universal solvable word problem group Suppose G were a universal solvable word problem group. Given a finite presentation P