# Higman's embedding theorem

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'altro canto, 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 (fino all'isomorfismo) 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.

Come corollario, there is a universal finitely presented group that contains all finitely presented groups as subgroups (fino all'isomorfismo); infatti, its finitely generated subgroups are exactly the finitely generated recursively presented groups (ancora, fino all'isomorfismo).

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. Infatti, 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. Serie A. Mathematical and Physical Sciences. vol. 262 (1961), pp. 455-475. ^ Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 2001. "I classici in matematica" series, reprint of the 1977 edition. ISBN 978-3-540-41158-1 Categorie: Infinite group theoryTheorems in group theory

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