Group isomorphism problem

{{Short description|Decision problem}}

{{Distinguish|Graph isomorphism problem}}

In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations refer to isomorphic groups.

The isomorphism problem was formulated by Max Dehn,{{cite journal |last=Dehn |first=Max |date=1911 |author-link=Max Dehn |title=Über unendliche diskontinuierliche Gruppenn |journal=Math. Ann. |volume=71 |pages=116–144 |doi=10.1007/BF01456932 | s2cid=123478582}} and together with the word problem and conjugacy problem, is one of three fundamental decision problems in group theory he identified in 1911.{{cite book |last1=Magnus |first1=Wilhelm |author-link1=Wilhelm Magnus |last2=Karrass |first2=Abraham |last3=Solitar |first3=Donald |author-link3=Donald Solitar |name-list-style=amp |date=1996 |title=Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations |edition=2nd |url=https://vdoc.pub/download/combinatorial-group-theory-presentations-of-groups-in-terms-of-generators-and-relations-1hi405kpbp4g |location=New York |publisher=Dover Publications |pages=24–29 |isbn=0486632814 |access-date=14 October 2022 |via=VDOC.PUB}} All three problems, formulated as ranging over all finitely presented groups, are undecidable. In the case of the isomorphism problem, this means that there does not exist a computer algorithm that takes two finite group presentations and decides whether or not the groups are isomorphic, regardless of how (finitely) much time is allowed for the algorithm to run and how (finitely) much memory is available. In fact the problem of deciding whether a finitely presented group is trivial is undecidable,{{cite book |last=Miller |first=Charles F. III |editor-last1=Baumslag |editor-first1=Gilbert |editor-last2=Miller |editor-first2=C. F. III |date=1992 |chapter=Decision Problems for Groups—survey and Reflections |chapter-url=https://link.springer.com/chapter/10.1007/978-1-4613-9730-4_1.pdf |chapter-url-access=subscription |title=Algorithms and Classification in Combinatorial Group Theory |url=https://link.springer.com/book/10.1007/978-1-4613-9730-4 |url-access=subscription |series=Mathematical Sciences Research Institute Publications |volume=23 |location=New York |publisher=Springer-Verlag |doi=10.1007/978-1-4613-9730-4_1 |isbn=9781461397328 |pages=1–59}} (See Corollary 3.4) a consequence of the Adian–Rabin theorem due to Sergei Adian and Michael O. Rabin.

However, there are some classes of finitely presented groups for which the restriction of the isomorphism problem is known to be decidable. They include finitely generated abelian groups, finite groups, Gromov-hyperbolic groups,{{cite journal |last1=Dahmani |first1=François |last2=Guirardel |first2=Vincent |year=2011 |title=The isomorphism problem for all hyperbolic groups |journal=Geometric and Functional Analysis |volume=21 |issue=2 |pages=223–300 |arxiv=1002.2590 |doi=10.1007/s00039-011-0120-0 |s2cid=115165062}} virtually torsion-free relatively hyperbolic groups with nilpotent parabolics,{{Cite journal |last1=Dahmani |first1=François |last2=Touikan |first2=Nicholas |date=2019-01-01 |title=Deciding isomorphy using Dehn fillings, the splitting case |url=https://link.springer.com/article/10.1007/s00222-018-0824-y |journal=Inventiones Mathematicae |language=en |volume=215 |issue=1 |pages=81–169 |doi=10.1007/s00222-018-0824-y |issn=1432-1297|arxiv=1311.3937 |bibcode=2019InMat.215...81D }} one-relator groups with non-trivial center,{{Cite journal |last=Pietrowski |first=Alfred |date=1974-06-01 |title=The isomorphism problem for one-relator groups with non-trivial centre |url=https://link.springer.com/article/10.1007/BF01214345 |journal=Mathematische Zeitschrift |language=en |volume=136 |issue=2 |pages=95–106 |doi=10.1007/BF01214345 |issn=1432-1823}} and two-generator one-relator groups with torsion.{{Cite journal |last=Pride |first=Stephen J. |date=1977 |title=The isomorphism problem for two-generator one-relator groups with torsion is solvable |url=https://www.ams.org/journals/tran/1977-227-00/S0002-9947-1977-0430085-X/ |journal=Transactions of the American Mathematical Society |language=en |volume=227 |pages=109–139 |doi=10.1090/S0002-9947-1977-0430085-X |issn=0002-9947}}

The group isomorphism problem, restricted to the groups that are given by multiplication tables, can be reduced to a graph isomorphism problem but not vice versa.{{Cite web |last=Hartnett |first=Kevin |date=23 June 2023 |title=Computer Scientists Inch Closer to Major Algorithmic Goal |url=https://www.quantamagazine.org/computer-scientists-inch-closer-to-major-algorithmic-goal-20230623/ |website=Quanta Magazine}} Both have quasi-polynomial-time algorithms, the former since 1978 attributed to Robert Tarjan{{Cite book |last=Miller |first=Gary L. |chapter=On the n^{log n} isomorphism technique (A Preliminary Report) |date=1978 |title=Proceedings of the tenth annual ACM symposium on Theory of computing - STOC '78 |chapter-url=http://portal.acm.org/citation.cfm?doid=800133.804331 |language=en |publisher=ACM Press |pages=51–58 |doi=10.1145/800133.804331|isbn=978-1-4503-7437-8 }} and the latter since 2015 by László Babai.{{citation |last=Babai |first=László |title=Graph isomorphism update |date=January 9, 2017 |url=http://people.cs.uchicago.edu/~laci/update.html}} A small but important improvement for the case p-groups of class 2 was obtained in 2023 by Xiaorui Sun.{{Cite arXiv |last=Sun |first=Xiaorui |date=2023 |title=Faster Isomorphism for p-Groups of Class 2 and Exponent p |class=cs.DS |eprint=2303.15412}}