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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.15497 |
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Table of 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.