Higman's embedding theorem



In 1961 G. Higman proved an important result in group theory that brings together in an intriguing way concepts from computability theory and finite presentability.

The result

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.


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. The importance of this conjecture 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).


  • G. Higman, Subgroups of finitely presented groups , Proc. Royal. Soc. London Ser. A (1961) pp.455-475.

  • L. Dediu , Higman’s Embedding theorem - An Elementary Proof , Report CDMCTS-010 (1995). (pdf)

  • W. F. Lawvere, On the effective topos - message to catlist, January 2002. (link)

Last revised on August 9, 2016 at 18:11:44. See the history of this page for a list of all contributions to it.