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Parallel Algorithms for Group Isomorphism via Code Equivalence

Author Michael Levet

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LIPIcs.SWAT.2026.30.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • Theory of computation → Parallel algorithms
  • Mathematics of computing → Combinatorial algorithms
Keywords
  • Group Isomorphism
  • Circuit Complexity
  • Code Equivalence

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Abstract

In this paper, we exhibit AC³ isomorphism tests for coprime extensions H ⋉ N where H is elementary Abelian and N is Abelian; and groups where Rad(G) = Z(G) is elementary Abelian and G = Soc^{*}(G). The fact that isomorphism testing for these families is in P was established respectively by Qiao, Sarma, and Tang (STACS 2011), and Grochow and Qiao (CCC 2014, SIAM J. Comput. 2017). 
The polynomial-time isomorphism tests for both of these families crucially leveraged small (size O(log |G|)) instances of Linear Code Equivalence (Babai, SODA 2011). Here, we combine Luks' group-theoretic method for Graph Isomorphism (FOCS 1980, J. Comput. Syst. Sci. 1982) with the fact that G is given by its multiplication table, to implement the corresponding instances of Linear Code Equivalence in AC³.
As a byproduct of our work, we show that isomorphism testing of arbitrary central-radical groups is decidable using AC circuits of depth O(log³ n) and size n^{O(log log n)}. This improves upon the previous bound of n^{O(log log n)}-time due to Grochow and Qiao (ibid.).

Cite As Get BibTex

Michael Levet. Parallel Algorithms for Group Isomorphism via Code Equivalence. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SWAT.2026.30

Author Details

Michael Levet
  • Department of Computer Science, College of Charleston, SC, USA

Funding

This work was partially supported by CRC 358 Integral Structures in Geometry and Number Theory at Bielefeld and Paderborn, Germany; the Department of Mathematics at Colorado State University; James B. Wilson’s NSF grant DMS-2319370; and travel support from the Department of Computer Science at the College of Charleston.

Acknowledgements

I wish to thank Peter Brooksbank, Edinah Gnang, Joshua A. Grochow, Takunari Miyazaki, and James B. Wilson for helpful discussions, as well as the anonymous referees for their helpful feedback. Parts of this work began at Tensors Algebra-Geometry-Applications (TAGA) 2024. I wish to thank Elina Robeva, Christopher Voll, and James B. Wilson for organizing this conference.

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