Subgroups of Finitely Presented Groups

TLDR: The main theorem of as discussed by the authors states that a finitely generated group can be embedded in a finite presented group if and only if it has a recursively enumerable set of defining relations.

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 A belian 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. A nother 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.