Higman’s embedding theorem.A finitely generated group can be embedded in a finitely presented group iff it has a presentation (with a finite generating set) for which the set of defining relations is a recursively enumerable set of words.

Theorem.There is a finitely presented group which contains an isomorphic copy of every finitely presented group.

Remark

It clearly makes sense to ask the same question for other algebraic theories. In fact, it has been conjectured by the group theorist W. Boone that a similar result holds more generally for single-sorted algebraic theories. While in fact the conjecture is not true for all single-sorted algebraic theories (see Boone conjecture for further details), as was known by Soviet mathematicians, the general question “In which algebraic categories does (an analogue of) Higman’s embedding theorem hold?” is still interesting. The importance of this question has been stressed and the parallel to results by Craig and Vaught in first-order logic has been pointed out by F. W. Lawvere (2002) (see at Boone conjecture for further details).