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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 finitely generated subgroups of finitely presented groups.
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Higman's embedding theorem also implies the
Novikov-Boone theorem (originally proved in the 1950s by other methods) about the existence of a
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finitely presented groups as subgroups (up to isomorphism); in fact, its finitely generated subgroups are exactly the finitely generated
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Proceedings of the Royal
Society. Series A. Mathematical and Physical Sciences. vol. 262 (1961), pp. 455-475.
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group is a subgroup of a finitely generated group, the theorem can be restated for those groups.
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Springer-Verlag, New York, 2001. "Classics in
Mathematics" series, reprint of the 1977 edition.
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112:which can be shown to have a finite presentation.
100:The usual proof of the theorem uses a sequence of
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130:Subgroups of finitely presented groups.
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76:universal finitely presented group
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93:with algorithmically undecidable
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86:(again, up to isomorphism).
84:recursively presented groups
150:Combinatorial Group Theory.
36:recursively presented group
18:Universal embedding theorem
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29:Higman's embedding theorem
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181:Theorems in group theory
108:and ending with a group
91:finitely presented group
47:finitely presented group
16:Not to be confused with
52:. This is a result of
176:Infinite group theory
41:can be embedded as a
33:finitely generated
31:states that every
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