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Algorithms for Finite Group Epimorphism Testing

Authors Joshua A. Grochow , Pranjal Srivastava , Dhara Thakkar

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LIPIcs.ICALP.2026.104.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithm design techniques
  • Theory of computation → Algebraic complexity theory
Keywords
  • Group epimorphism problem
  • group-theoretic algorithms
  • polynomial-time algorithms
  • normal Hall subgroups

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Abstract

The Group Epimorphism Problem (GpEpi) asks, given two finite groups G₁ and G₂, whether there exists a surjective group homomorphism, or epimorphism, from G₁ to G₂. When the input groups are given by their multiplication (Cayley) tables, the problem admits a quasipolynomial-time algorithm in general, but little is known about its complexity for structured classes of finite groups. In this paper, we study the computational complexity of GpEpi for several well-studied classes of finite groups.
Our main results are polynomial-time epimorphism tests for several classes of groups for which polynomial-time isomorphism testing was previously known:  
- Groups with Abelian normal Hall subgroups with cyclic complement
- Groups with (product of) elementary Abelian normal Hall subgroup with elementary Abelian complement.
- Groups with some constraints on their Abelian chief factors.

Cite As Get BibTex

Joshua A. Grochow, Pranjal Srivastava, and Dhara Thakkar. Algorithms for Finite Group Epimorphism Testing. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 104:1-104:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.104

Author Details

Joshua A. Grochow
  • Department of Computer Science, University of Colorado Boulder, CO, USA
  • Department of Mathematics, University of Colorado Boulder, CO, USA
Pranjal Srivastava
  • Department of Maths and Computing, Indian Institute of Information Technology Vadodara, Gandhinagar, India
Dhara Thakkar
  • Graduate School of Mathematics, Nagoya University, Japan

Funding

  • Grochow, Joshua A.: This author was supported by NSF CAREER award CCF-2047756.
  • Thakkar, Dhara: The author is supported by JSPS KAKENHI 24H00071 and 25K24674.

Acknowledgements

We thank the ICALP reviewers for helpful comments on the submission. DT would also like to thank Armin Weiss for bringing the Group Epimorphism problem to her attention.

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