Abstract
Free groups are central to group theory, and are ubiquitous across many branches of mathematics, including algebra, topology and geometry. An important result in the theory of free groups is the Nielsen-Schreier Theorem, which states that any subgroup of a free group is free. In this paper, we present a formalisation, in Isabelle/HOL, of a combinatorial proof of the Nielsen-Schreier theorem. In particular, our formalisation applies to arbitrary subgroups of free groups, without any restriction on the index of the subgroup or the cardinality of its generating sets. We also present a formalisation of an algorithm which determines whether two group words represent conjugate elements in a free group.
To the best of our knowledge, our work is the first formalisation of a combinatorial proof of the Nielsen-Schreier theorem in any proof assistant; the first formalisation of a proof of the Nielsen-Schreier theorem in Isabelle/HOL; and the first formalisation of the decision process for the conjugacy problem for free groups in any proof assistant.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aransay, J., et al.: The Isabelle/HOL Algebra Library (2022). https://isabelle.in.tum.de/library/HOL/HOL-Algebra/. Accessed 11 June 2023
Baer, R., Levi, F.: Freie Produkte und ihre Untergruppen. Compos. Math. 3, 391–398 (1936)
Bapanapally, J., Gamboa, R.: A complete, mechanically-verified proof of the Banach-Tarski theorem in ACL2(R). In: Andronick, J., de Moura, L. (eds.) 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), vol. 237, pp. 5:1–5:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2022). https://doi.org/10.4230/LIPIcs.ITP.2022.5. https://drops.dagstuhl.de/opus/volltexte/2022/16714
Breitner, J.: Free groups. Archive of Formal Proofs (2010). https://isa-afp.org/entries/Free-Groups.html. Formal proof development
Dehn, M.: Über unendliche diskontinuierliche Gruppen. Math. Ann. 71(1), 116–144 (1911)
Dyck, W.: Gruppentheoretische studien. Math. Ann. 20(1), 1–44 (1882)
Holub, Š., Starosta, Š.: Formalization of basic combinatorics on words. In: Cohen, L., Kaliszyk, C. (eds.) 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), vol. 193, pp. 22:1–22:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2021). https://doi.org/10.4230/LIPIcs.ITP.2021.22. https://drops.dagstuhl.de/opus/volltexte/2021/13917
JRH13: Simple formulation of group theory with a type of “(A)group”. GitHub repository. https://github.com/jrh13/hol-light/blob/master/Library/grouptheory.ml
Kharlampovich, O., Myasnikov, A.: Elementary theory of free non-abelian groups. J. Algebra 302(2), 451–552 (2006)
Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory, vol. 188. Springer, Heidelberg (1977)
mathlib: Free Groups - Lean Mathematical Library. https://leanprover-community.github.io/mathlib_docs/group_theory/free_group.html. Accessed 09 Apr 2023
mathlib: The Nielsen-Schreier theorem - Lean Mathematical Library. https://leanprover-community.github.io/mathlib_docs/group_theory/nielsen_schreier.html. Accessed 02 June 2023
Nielsen, J.: Om Regning med ikke-kommutative Faktorer og dens Anvendelse i Gruppeteorien. Matematisk Tidsskrift. B 77–94 (1921)
Schepler, D.: Freegroups Coq contribution. GitHub repository (2011). https://github.com/coq-contribs/free-groups
Schreier, O.: Die untergruppen der freien gruppen. Abhandlungen aus dem Mathematischen Seminar der universität Hamburg 5, 161–183 (1927)
Sela, Z.: Diophantine geometry over groups VI: the elementary theory of a free group. Geom. Funct. Anal. 16(3), 707–730 (2006)
Swan, A.W.: On the Nielsen-Schreier theorem in homotopy type theory. Log. Methods Comput. Sci. 18 (2022)
Acknowledgements
This work was supported by the Krea Faculty Research Fellowship “Computational Thought in Group Theory and Geometry”. In addition, we are grateful to the annonymous reviewers for their throughful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Kharim, A.S.A., Prathamesh, T.V.H., Rajiv, S., Vyas, R. (2023). Formalizing Free Groups in Isabelle/HOL: The Nielsen-Schreier Theorem and the Conjugacy Problem. In: Dubois, C., Kerber, M. (eds) Intelligent Computer Mathematics. CICM 2023. Lecture Notes in Computer Science(), vol 14101. Springer, Cham. https://doi.org/10.1007/978-3-031-42753-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-031-42753-4_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-42752-7
Online ISBN: 978-3-031-42753-4
eBook Packages: Computer ScienceComputer Science (R0)