Higman's embedding theorem

In group theory, Higman's embedding theorem states that every finitely generated group whose relations are recursively enumerated can be embedded as a subgroup of some finitely presented group.

Since every countably generated group is a subgroup of a finitely generated group, the theorem can be restated for those groups.

As a corollary, there is a universal finitely presented group that contains all finitely presented groups as subgroups (up to isomorphism); in fact, its finitely generated subgroups are exactly the finitely generated recursively presented groups (again, up to isomorphism).

The usual proof of the theorem uses HNN extensions.