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Subgroups of Finitely Presented Groups

G. Higman
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 262, No. 1311 (Aug. 8, 1961), pp. 455-475
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/2414348
Page Count: 21
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Subgroups of Finitely Presented Groups
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Abstract

The main theorem of this paper states that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. It follows that every countable Abelian group, and every countable locally finite group can be so embedded; and that there exists a finitely presented group which simultaneously embeds all finitely presented groups. Another corollary of the theorem is the known fact that there exist finitely presented groups with recursively insoluble word problem. A by-product of the proof is a genetic characterization of the recursively enumerable subsets of a suitable effectively enumerable set.

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