# A consequence from a proof of the Higman Embedding Theorem

*by Maurice Chiodo*

*Institution:*Mathematics and Statisics, University of Melbourne

*Date: Thu 28th May 2009*

*Time: 4:15 PM*

*Location: Room 107, Richard Berry Building, Uni of Melb*

*Abstract*: The Higman embedding theorem states that a finitely generated group

can be embedded into a finitely presented group iff it is recursively

presented. Using parts of a proof of this theorem, as well as a

theorem by Rabin, we shall show the following:

There exists a recursive list of (recursively constructible) finite

presentations of groups P_{1}, P_{2}, ... such that, if i=

Cantor pairing function of m and n) then P_{i} is isomorphic to the

trivial group iff the turing machine T_{m} halts on input N (ie: n \in

W_{m}).

Only the necessary parts of the Higman embedding theorem will be used,

and these will be stated without full proof.

*For More Information:* For more information please contact: G.Hjorth@ms.unimelb.edu.au