Subgroups of finitely presented groups

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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