# nLab Higman's embedding theorem

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

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.

## 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. 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).

## References

• 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.