Polynomial-time isomorphism test for $k$-generated extensions of abelian groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915841634729984 |
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| author | Skresanov, Saveliy V. |
| author_facet | Skresanov, Saveliy V. |
| contents | The group isomorphism problem asks whether two finite groups given by their Cayley tables are isomorphic or not. Although there are polynomial-time algorithms for some specific group classes, the best known algorithm for testing isomorphism of arbitrary groups of order $ n $ has time complexity $ n^{O(\log n)} $. We consider the group isomorphism problem for some extensions of abelian groups by $ k $-generated groups for bounded $ k $. In particular, we prove that one can test isomorphism of abelian-by-cyclic extensions in polynomial time, generalizing a 2009 result of Le Gall for coprime extensions. As another application, we give a polynomial-time isomorphism test for abelian-by-simple group extensions, generalizing a 2017 result of Grochow and Qiao for central extensions. The main novelty of the proof is a polynomial-time algorithm for computing the unit group of a finite ring, which might be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_15497 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Polynomial-time isomorphism test for $k$-generated extensions of abelian groups Skresanov, Saveliy V. Group Theory Computational Complexity 20-08 (Primary) 68Q25, 16Z05 (Secondary) The group isomorphism problem asks whether two finite groups given by their Cayley tables are isomorphic or not. Although there are polynomial-time algorithms for some specific group classes, the best known algorithm for testing isomorphism of arbitrary groups of order $ n $ has time complexity $ n^{O(\log n)} $. We consider the group isomorphism problem for some extensions of abelian groups by $ k $-generated groups for bounded $ k $. In particular, we prove that one can test isomorphism of abelian-by-cyclic extensions in polynomial time, generalizing a 2009 result of Le Gall for coprime extensions. As another application, we give a polynomial-time isomorphism test for abelian-by-simple group extensions, generalizing a 2017 result of Grochow and Qiao for central extensions. The main novelty of the proof is a polynomial-time algorithm for computing the unit group of a finite ring, which might be of independent interest. |
| title | Polynomial-time isomorphism test for $k$-generated extensions of abelian groups |
| topic | Group Theory Computational Complexity 20-08 (Primary) 68Q25, 16Z05 (Secondary) |
| url | https://arxiv.org/abs/2602.15497 |