= Solution
The <Higman embedding theorem> says that a <finitely generated group> embeds in a <finitely presented group> if and only if it has a <recursive presentation of a group>, that is, a presentation on finitely many generators with a recursively enumerable set of relators. We use its embedding direction as an assumed theorem. The reverse direction follows by expressing subgroup generators as words in the larger group and enumerating finite products of conjugates of its relators and their inverses; comparison after free reduction enumerates all null words in the subgroup. This is the <enumerable identities of finitely generated subgroups> argument. The theorem does not assume that the recursively presented group's <word problem for a group> is decidable.
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