A finitely generated group embeds in a finitely presented group if and only if it has a recursive presentation of a group. The necessity follows by enumerating proofs that substituted generator words are trivial in the larger finite group presentation. The sufficiency is the embedding theorem; it also applies to effectively countably generated recursive presentations after an effective two-generator embedding. This is the theorem of Subgroups of finitely presented groups.
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Higman's embedding theorem is a result in the field of formal languages and automata theory, specifically relating to the study of recursively enumerable languages and context-free languages. The theorem provides a way to understand the structure of certain algebraic objects associated with these languages.