Higman's Embedding Theorem

Higman's Embedding Theorem, proved by Graham Higman in the 1960s, states that every finitely generated recursively presented group can be embedded as a subgroup of some finitely presented group. Since every finitely generated subgroup of a finitely presented group is itself recursively presented, the theorem shows that the finitely generated recursively presented groups are, up to isomorphism, exactly the finitely generated subgroups of finitely presented groups, and it implies the earlier Novikov-Boone theorem on the existence of a finitely presented group with an algorithmically undecidable word problem. The usual proof builds a sequence of HNN extensions starting from the given group and ending in a group that has a finite presentation.