Embedding in f.p. simple groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:59:38Z http://mathoverflow.net/feeds/question/73568 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73568/embedding-in-f-p-simple-groups Embedding in f.p. simple groups Victor 2011-08-24T14:51:10Z 2011-08-24T16:56:11Z <p>Dear All!</p> <p>At the time when Lyndon and Schupp wrote their book there was an open question:</p> <p>Question: Does every finitely presented group with soluble word problem embed in a finitely presented simple group?</p> <p>Is it still open? Could you hint at some useful references about this? Thanks!</p> http://mathoverflow.net/questions/73568/embedding-in-f-p-simple-groups/73585#73585 Answer by Derek Holt for Embedding in f.p. simple groups Derek Holt 2011-08-24T16:56:11Z 2011-08-24T16:56:11Z <p>I believe it is still open. By the Boone-Higman Theorem (W. W. Boone and G. Higman, "An algebraic characterization of the solvability of the word problem", J. Austral. Math. Soc. 18, 41-53 (1974)), a finitely presented group has solvable word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group.</p> <p>It is widely believed that it is possible for the simple group itself to be finitely presented, but (AFAIK) not proved.</p> <p>So the answer to Agol's comment is that no finitely presented group with unsolvable word problem can be embedded into a finitely presented simple group.</p>