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Polynomial-time isomorphism test for $k$-generated extensions of abelian groups

Saveliy V. Skresanov

TL;DR

The paper tackles the group isomorphism problem for noncoprime extensions of abelian groups by $k$-generated groups, achieving polynomial-time isomorphism tests for abelian-by-cyclic and abelian-by-simple extensions. The central technique is a polynomial-time computation of the unit group $R^ imes$ of finite rings acting on a finite abelian group, enabling a reduction of extension isomorphism to compatible actions and cocycle data, plus a coset-of-isomorphisms computation. Together with a tower-based approach for extensions, these results extend tractable cases beyond coprime settings and broaden the classes of groups for which isomorphism testing can be done efficiently. The work has potential impact on computational group theory and related areas by providing new structural tools to handle noncoprime extensions and by offering a polynomial-time path to compute automorphism cosets in these contexts.

Abstract

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.

Polynomial-time isomorphism test for $k$-generated extensions of abelian groups

TL;DR

The paper tackles the group isomorphism problem for noncoprime extensions of abelian groups by -generated groups, achieving polynomial-time isomorphism tests for abelian-by-cyclic and abelian-by-simple extensions. The central technique is a polynomial-time computation of the unit group of finite rings acting on a finite abelian group, enabling a reduction of extension isomorphism to compatible actions and cocycle data, plus a coset-of-isomorphisms computation. Together with a tower-based approach for extensions, these results extend tractable cases beyond coprime settings and broaden the classes of groups for which isomorphism testing can be done efficiently. The work has potential impact on computational group theory and related areas by providing new structural tools to handle noncoprime extensions and by offering a polynomial-time path to compute automorphism cosets in these contexts.

Abstract

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 has time complexity . We consider the group isomorphism problem for some extensions of abelian groups by -generated groups for bounded . 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.
Paper Structure (9 sections, 15 theorems, 21 equations)

This paper contains 9 sections, 15 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Isomorphisms of extensions
  4. Computing unit groups of finite rings
  5. Proof of Theorem \ref{['main']}
  6. Finding an isomorphism
  7. Computing the coset of isomorphisms
  8. Proof of Theorem \ref{['ctower']}
  9. Acknowledgements

Key Result

Theorem 1.1

Let $G$ and $G_0$ be finite groups of order $n$ given by their Cayley tables. Suppose we are also given abelian normal subgroups $A \unlhd G$ and $A_0 \unlhd G_0$, and an isomorphism $\psi : G/A \to G_0/A_0$. If $G/A$ is $k$-generated, then we can test in time polynomial in $n^k$ whether there exist

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Proposition 2.1: iuliana and iuliana-thesis
  • Proposition 2.2: dio1dio2
  • Theorem 3.1
  • Lemma 4.1
  • ...and 11 more