Théorème d'intégration de Higman

Higman's embedding theorem Not to be confused with Universal embedding theorem.

In group theory, Higman's embedding theorem states that every finitely generated recursively presented group R can be embedded as a subgroup of some finitely presented group G. This is a result of Graham Higman from the 1960s.[1] D'autre part, it is an easy theorem that every finitely generated subgroup of a finitely presented group is recursively presented, so the recursively presented finitely generated groups are (jusqu'à l'isomorphisme) exactly the finitely generated subgroups of finitely presented groups.

Since every countable group is a subgroup of a finitely generated group, the theorem can be restated for those groups.

As a corollary, there is a universal finitely presented group that contains all finitely presented groups as subgroups (jusqu'à l'isomorphisme); En fait, its finitely generated subgroups are exactly the finitely generated recursively presented groups (encore, jusqu'à l'isomorphisme).

Higman's embedding theorem also implies the Novikov-Boone theorem (originally proved in the 1950s by other methods) about the existence of a finitely presented group with algorithmically undecidable word problem. En effet, it is fairly easy to construct a finitely generated recursively presented group with undecidable word problem. Then any finitely presented group that contains this group as a subgroup will have undecidable word problem as well.

The usual proof of the theorem uses a sequence of HNN extensions starting with R and ending with a group G which can be shown to have a finite presentation.[2] References ^ Graham Higman, Subgroups of finitely presented groups. Proceedings of the Royal Society. Série A. Mathematical and Physical Sciences. volume. 262 (1961), pp. 455-475. ^ Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer Verlag, New York, 2001. "Classiques en mathématiques" series, reprint of the 1977 edition. ISBN 978-3-540-41158-1 Catégories: Infinite group theoryTheorems in group theory