Published September 12, 2026
| Version 1.0.0
Preprint
Open
HNN extensions and inner homogeneity of group representations over commutative rings
Description
Let \(R\) be a nonzero commutative ring. Every \(R\)-representation \((G,V)\) embeds in \((\Gamma,W)\) with \[\ker\bigl(\Gamma\to\operatorname{Aut}_R(W)\bigr)=1,\qquad |\Gamma|,|W|\le\max\{\aleph_0,|R|,|G|,|V|\},\] such that every isomorphism between finitely generated subrepresentations is induced by \((g\mapsto tgt^{-1},\,w\mapsto tw)\) for some \(t\in\Gamma\). A local HNN construction realizes arbitrary subrepresentation isomorphisms by adjoining the corresponding linear relations to an induced module. Injectivity follows from the boundary map of a tree of modules, with no splitting or projectivity assumption. The field case answers Kourovka Problem 17.101.
Files
kourovka-17-101-paper.pdf
Additional details
Software
- Repository URL
- https://github.com/Achxy/k-17-101
- Programming language
- Lean
- Development Status
- Active
References
- E. V. Aladova, A. A. Gvaramiya and B. I. Plotkin, Logic in representations of groups, Algebra and Logic 51 (2012), 1–27. https://doi.org/10.1007/s10469-012-9167-8 Russian original: Algebra i Logika 51 (2012), 3–40. https://www.mathnet.ru/al520
- W. Dicks, Mayer–Vietoris presentations over colimits of rings, Proc. London Math. Soc. (3) 34 (1977), no. 3, 557–576. https://doi.org/10.1112/plms/s3-34.3.557
- W. Dicks, The HNN construction for rings, J. Algebra 81 (1983), no. 2, 434–487. https://doi.org/10.1016/0021-8693(83)90198-9
- G. Higman, B. H. Neumann and H. Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), no. 4, 247–254. https://doi.org/10.1112/jlms/s1-24.4.247
- O. H. Kegel, Examples of highly transitive permutation groups, Rend. Sem. Mat. Univ. Padova 63 (1980), 295–300.
- E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st ed., Problem 17.101. https://arxiv.org/abs/1401.0300v46