Group isomorphism problem explained

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,^[1] and together with the word problem and conjugacy problem, is one of three fundamental decision problems in group theory he identified in 1911.^[2] All three problems are undecidable: there does not exist a computer algorithm that correctly solves every instance of the isomorphism problem, or of the other two problems, regardless of how much time is allowed for the algorithm to run. In fact the problem of deciding whether a group is trivial is undecidable,^[3] a consequence of the Adian–Rabin theorem due to Sergei Adian and Michael O. Rabin.

The group isomorphism problem, in which the groups are given by multiplication tables, can be reduced to a graph isomorphism problem but not vice versa.^[4] Both have quasi-polynomial-time algorithms, the former since 1978 attributed to Robert Tarjan^[5] and the latter since 2015 by László Babai. A small but important improvement for the case p-groups of class 2 was obtained in 2023 by Xiaorui Sun.^[6]

References

• Book: Johnson, D. L. . 1997 . Presentations of Groups . 2nd . Cambridge . . 10.1017/CBO9781139168410 . 0521372038 . 49.

Notes and References

1. Dehn . Max . 1911 . Max Dehn . Über unendliche diskontinuierliche Gruppenn . . 71 . 116–144 . 10.1007/BF01456932 . 123478582.
2. Book: Magnus . Wilhelm . Wilhelm Magnus . Karrass . Abraham . Solitar . Donald . Donald Solitar . amp . 1996 . Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations . 2nd . New York . . 24–29 . 0486632814 . 14 October 2022 . VDOC.PUB.
3. Book: Miller, Charles F. III . Baumslag . Gilbert . Miller . C. F. III . 1992 . Decision Problems for Groups—survey and Reflections . https://link.springer.com/chapter/10.1007/978-1-4613-9730-4_1.pdf . subscription . Algorithms and Classification in Combinatorial Group Theory . subscription . Mathematical Sciences Research Institute Publications . 23 . New York . Springer-Verlag . 10.1007/978-1-4613-9730-4_1 . 9781461397328 . 1–59. (See Corollary 3.4)
4. Web site: Hartnett . Kevin . 23 June 2023 . Computer Scientists Inch Closer to Major Algorithmic Goal . Quanta Magazine.
5. Book: Miller, Gary L. . On the n^{log n} isomorphism technique (A Preliminary Report) . 1978 . Proceedings of the tenth annual ACM symposium on Theory of computing - STOC '78 . http://portal.acm.org/citation.cfm?doid=800133.804331 . en . ACM Press . 51–58 . 10.1145/800133.804331. 978-1-4503-7437-8 .
6. Sun . Xiaorui . 2023 . Faster Isomorphism for p-Groups of Class 2 and Exponent p . cs.DS . 2303.15412.