TY - GEN
T1 - Formalizing Free Groups in Isabelle/HOL
T2 - Proceedings of the 16th Conference on Intelligent Computer Mathematics, CICM 2023
AU - Kharim, Aabid Seeyal Abdul
AU - Prathamesh, T. V.H.
AU - Rajiv, Shweta
AU - Vyas, Rishi
N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.
PY - 2023/1/1
Y1 - 2023/1/1
N2 - 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.
AB - 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.
KW - Isabelle/HOL
KW - free groups
KW - group theory
UR - http://www.scopus.com/inward/record.url?scp=85172081379&partnerID=8YFLogxK
U2 - 10.1007/978-3-031-42753-4_11
DO - 10.1007/978-3-031-42753-4_11
M3 - Conference contribution
AN - SCOPUS:85172081379
SN - 9783031427527
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 158
EP - 173
BT - Intelligent Computer Mathematics - 16th International Conference, CICM 2023, Proceedings
A2 - Dubois, Catherine
A2 - Kerber, Manfred
PB - Springer Science and Business Media Deutschland GmbH
Y2 - 5 September 2023 through 8 September 2023
ER -