On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors

Authors Joshua A. Grochow , Youming Qiao

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Joshua A. Grochow
  • Departments of Computer Science and Mathematics, University of Colorado Boulder, CO, USA
Youming Qiao
  • Centre for Quantum Software and Information, University of Technology Sydney, Australia


The authors would like to thank James B. Wilson for related discussions, and Ryan Williams for pointing out the problem of distinguishing between ETH and #ETH. J. A. G. would like to thank V. Futorny and V. V. Sergeichuk for their collaboration on the related work (Futorny, Grochow, and Sergeichuk, Lin. Alg. Appl., 2019). Ideas leading to this work originated from the 2015 workshop "Wildness in computer science, physics, and mathematics" at the Santa Fe Institute.

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Joshua A. Grochow and Youming Qiao. On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 16:1-16:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In this paper we study some classical complexity-theoretic questions regarding Group Isomorphism (GpI). We focus on p-groups (groups of prime power order) with odd p, which are believed to be a bottleneck case for GpI, and work in the model of matrix groups over finite fields. Our main results are as follows. - Although search-to-decision and counting-to-decision reductions have been known for over four decades for Graph Isomorphism (GI), they had remained open for GpI, explicitly asked by Arvind & Torán (Bull. EATCS, 2005). Extending methods from Tensor Isomorphism (Grochow & Qiao, ITCS 2021), we show moderately exponential-time such reductions within p-groups of class 2 and exponent p. - Despite the widely held belief that p-groups of class 2 and exponent p are the hardest cases of GpI, there was no reduction to these groups from any larger class of groups. Again using methods from Tensor Isomorphism (ibid.), we show the first such reduction, namely from isomorphism testing of p-groups of "small" class and exponent p to those of class two and exponent p. For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of GI. Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor & Lubotzky, Geom. Dedicata, 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard Correspondence with Tensor Isomorphism-completeness results (Grochow & Qiao, ibid.).

Subject Classification

ACM Subject Classification
  • Computing methodologies → Algebraic algorithms
  • Theory of computation → Problems, reductions and completeness
  • group isomorphism
  • search-to-decision reduction
  • counting-to-decision reduction
  • nilpotent group isomorphism
  • p-group isomorphism
  • tensor isomorphism


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