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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.〔 Graham Higman, ''Subgroups of finitely presented groups.'' Proceedings of the Royal Society. Series A. Mathematical and Physical Sciences. vol. 262 (1961), pp. 455-475.〕 On the other hand, 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 (up to isomorphism) exactly the 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 (up to isomorphism); in fact, its finitely generated subgroups are exactly the finitely generated recursively presented groups (again, up to isomorphism). 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. Indeed, 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.〔Roger C. Lyndon and Paul E. Schupp. ''Combinatorial Group Theory.'' Springer-Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition. ISBN 978-3-540-41158-1 〕 ==References== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Higman's embedding theorem」の詳細全文を読む スポンサード リンク
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