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 decide 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.

翻译：群同构问题旨在判定由凯莱表给出的两个有限群是否同构。尽管针对某些特定群类存在多项式时间算法，但测试任意$n$阶群同构性的最佳已知算法时间复杂度为$n^{O(\log n)}$。本文研究有界$k$条件下阿贝尔群经$k$生成群扩张的同构问题。特别地，我们证明了阿贝尔群经循环群扩张的同构性可在多项式时间内判定，这推广了Le Gall于2009年关于互素扩张的结果。作为另一应用，我们给出了阿贝尔群经单群扩张的多项式时间同构判定算法，推广了Grochow和Qiao于2017年关于中心扩张的结果。证明的核心创新在于给出了计算有限环单位群的多项式时间算法，该算法可能具有独立的研究价值。