A relativized embedding theorem for recursively enumerable group presentations
For every oracle $X\subseteq\mathbb N$ we construct a three-generated group $G_X$ with an $X$-computably enumerable presentation such that a finitely generated group has an $X$-computably enumerable presentation if and only if it embeds as a finitely generated subgroup of a quotient of a finite free product of copies of $G_X$ by finitely many relators. This gives an affirmative answer to Kourovka Problem 20.33.
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Lemma 2.1
Every countably generated $X$-presented group embeds in a three-generated $X$-presented group.
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- External referenceG. Higman, Subgroups of finitely presented groups, Proc. Roy. Soc. London Ser. A 262 (1961), 455--475. DOI: 10.1098/rspa.1961.0132.LaTeX source\bibitem{higman} G. Higman, \emph{Subgroups of finitely presented groups}, Proc. Roy. Soc. London Ser. A \textbf{262} (1961), 455--475. DOI: 10.1098/rspa.1961.0132.
- External referenceG. Higman, B. H. Neumann and H. Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247--254. DOI: 10.1112/jlms/s1-24.4.247.LaTeX source\bibitem{hnn} G. Higman, B. H. Neumann and H. Neumann, \emph{Embedding theorems for groups}, J. London Math. Soc. \textbf{24} (1949), 247--254. DOI: 10.1112/jlms/s1-24.4.247.
- External referenceE. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, No. 21, Novosibirsk, 2026; arXiv:1401.0300, Problem 20.33.LaTeX source\bibitem{kourovka} E. I. Khukhro and V. D. Mazurov (eds.), \emph{Unsolved Problems in Group Theory: The Kourovka Notebook}, No. 21, Novosibirsk, 2026; arXiv:1401.0300, Problem 20.33.
- External referenceM. Chiodo and Z. McKenzie, $X$-torsion and universal groups, arXiv:1610.00313 (2016).LaTeX source\bibitem{chiodo-mckenzie} M. Chiodo and Z. McKenzie, \emph{$X$-torsion and universal groups}, arXiv:1610.00313 (2016).
- External referenceV. H. Mikaelian, Embeddings determined by universal words in the rank 2 free group, Siberian Math. J. 62 (2021), no. 1, 123--130. DOI: 10.1134/S0037446621010134; arXiv:2002.09433.LaTeX source\bibitem{mikaelian} V. H. Mikaelian, \emph{Embeddings determined by universal words in the rank 2 free group}, Siberian Math. J. \textbf{62} (2021), no. 1, 123--130. DOI: 10.1134/S0037446621010134; arXiv:2002.09433.
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Machine-readable theorem index · 2 statements
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- PF:2026.000016/v1/LEM-2.1
Every countably generated $X$-presented group embeds in a three-generated $X$-presented group.
- PF:2026.000016/v1/THM-3.1
For every $X\subseteq\mathbb N$ there is a three-generated $X$-presented group $G_X$ such that a finitely generated group $H$ is $X$-presented if and only if $H$ embeds as a finitely generated subgroup of \[ Q=\left(G_X^{(1)}*\cdots*G_X^{(m)}\right)/\langle\!\langle S\rangle\!\rangle \] for some finite $m$ and some finite set $S$.