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arxiv: 2604.13953 · v1 · submitted 2026-04-15 · 💻 cs.CC · cs.DS· math.GR

Parallel Algorithms for Group Isomorphism via Code Equivalence

Michael Levet This is my paper

Pith reviewed 2026-05-10 11:45 UTC · model grok-4.3

classification 💻 cs.CC cs.DSmath.GR
keywords group isomorphismlinear code equivalenceAC³ circuitsparallel algorithmscentral-radical groupscoprime extensionselementary abelian groups
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0 comments X

The pith

Isomorphism testing for coprime abelian extensions and certain central-radical groups is in AC³.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that two families of group isomorphism problems can be decided by AC³ circuits: coprime extensions H ⋉ N with H elementary abelian and N abelian, plus groups whose radical equals the center and is elementary abelian with the group equal to its socle star. It reaches this by reducing isomorphism to small linear code equivalence instances and solving those instances in AC³, using the fact that the groups are given explicitly by multiplication tables. Prior work already placed both families in polynomial time, but the new algorithms place them in a constant-depth parallel class. As a direct consequence, isomorphism testing for arbitrary central-radical groups falls into AC circuits of depth O(log³ n) and size n to the O(log log n).

Core claim

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 polynomial-time isomorphism tests for both of these families crucially leveraged small instances of Linear Code Equivalence. Here, we combine Luks' group-theoretic method for Graph Isomorphism 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, isomorphism testing of arbitrary central-radical groups is decidable using AC circuits of depth O(log³ n) and size n^{O(log log n)}.

What carries the argument

Small instances of linear code equivalence, solved in AC³ by applying Luks' group-theoretic techniques directly to the multiplication table presentation of the input groups.

If this is right

  • Isomorphism testing for the two specified families of groups lies in AC³.
  • Isomorphism testing for arbitrary central-radical groups lies in AC circuits of depth O(log³ n) and size n^{O(log log n)}.
  • The earlier polynomial-time algorithms for these families are strengthened to constant-depth parallel algorithms.
  • The reductions from group isomorphism to linear code equivalence can be realized inside AC³ when the group is presented by a multiplication table.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction-plus-Luks approach may yield AC³ or low-depth AC algorithms for additional families of groups whose isomorphism reduces to code equivalence.
  • Multiplication-table presentations appear especially well-suited for extracting high parallelism from algebraic decision problems.
  • The result supplies a concrete route to study the parallel complexity of graph isomorphism through its group-theoretic formulation.

Load-bearing premise

The input groups are given by their complete multiplication tables, which supplies direct access to the group operation inside the circuit.

What would settle it

An explicit pair of groups from one of the families whose isomorphism status cannot be correctly decided by any family of AC³ circuits built from the derived code-equivalence instances.

read the original abstract

In this paper, we exhibit $\textsf{AC}^{3}$ isomorphism tests for coprime extensions $H \ltimes N$ where $H$ is elementary Abelian and $N$ is Abelian; and groups where $\text{Rad}(G) = Z(G)$ is elementary Abelian and $G = \text{Soc}^{*}(G)$. The fact that isomorphism testing for these families is in $\textsf{P}$ 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 $\textsf{AC}^{3}$. As a byproduct of our work, we show that isomorphism testing of arbitrary central-radical groups is decidable using $\textsf{AC}$ circuits of depth $O(\log^3 n)$ 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.).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. In this paper, the authors exhibit AC³ isomorphism tests for coprime extensions H ⋉ N where H is elementary Abelian and N is Abelian, as well as for groups where Rad(G) = Z(G) is elementary Abelian and G = Soc*(G). They achieve this by implementing small instances of Linear Code Equivalence in AC³ circuits using Luks' group-theoretic method combined with the multiplication-table model of the groups. As a byproduct, they provide an AC circuit algorithm of depth O(log³ n) and size n^{O(log log n)} for isomorphism testing of arbitrary central-radical groups, improving upon previous time bounds.

Significance. If the central claims hold, this work is significant for advancing the parallel complexity theory of group isomorphism. It provides the first AC³ algorithms for these specific group families by parallelizing the code equivalence subroutines. The improvement to AC circuits with polylog depth for central-radical groups is a clear strengthening of prior results. The paper credits the polynomial-time foundations from Qiao-Sarma-Tang and Grochow-Qiao, and focuses on the parallel implementation, which is a valuable contribution in the field.

major comments (1)
  1. [the description of the AC³ implementation for Linear Code Equivalence (via Luks' method)] The claim that the small (O(log |G|)) Linear Code Equivalence instances arising in the isomorphism tests for the two families can be solved in AC³ via Luks' method depends on a non-trivial parallelization that exploits the multiplication table to avoid full enumeration. The manuscript must explicitly verify that computing stabilizers, testing conjugacy, and constructing the required linear transformations introduces no extra logarithmic depth layer or implicit sequential search, as any such step would violate the depth-3 bound even if the algorithm remains polynomial-time.
minor comments (1)
  1. [Abstract] The abstract and introduction should more explicitly contrast the new AC³ bound with the prior polynomial-time algorithms of Qiao-Sarma-Tang and Grochow-Qiao, including a brief remark on why the multiplication-table presentation is essential for the circuit construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the significance of our results and for the constructive major comment. We address the point regarding explicit verification of the AC³ implementation below.

read point-by-point responses

  1. Referee: The claim that the small (O(log |G|)) Linear Code Equivalence instances arising in the isomorphism tests for the two families can be solved in AC³ via Luks' method depends on a non-trivial parallelization that exploits the multiplication table to avoid full enumeration. The manuscript must explicitly verify that computing stabilizers, testing conjugacy, and constructing the required linear transformations introduces no extra logarithmic depth layer or implicit sequential search, as any such step would violate the depth-3 bound even if the algorithm remains polynomial-time.

    Authors: We agree that the manuscript would benefit from an explicit depth analysis to confirm the AC³ bound. In the revised version we will add a dedicated subsection detailing the parallelization. Because the Linear Code Equivalence instances have size O(log |G|), the multiplication-table representation permits constant-depth element access. Stabilizer and conjugacy computations are performed via parallel orbit-stabilizer algorithms whose depth is O(log log n) for these small instances; constructing the linear transformations reduces to parallel linear algebra over GF(q), which lies in AC¹. These factors compose with the existing O(log³ n) bound without introducing an additional logarithmic layer, keeping the overall algorithm in AC³. We will include the corresponding circuit-depth calculations and pseudocode. revision: yes

Circularity Check

0 steps flagged

No circularity: new AC^3 implementation of known small-code-equivalence instances

full rationale

The derivation rests on external prior results (Qiao-Sarma-Tang STACS 2011 and Grochow-Qiao CCC 2014) that already placed the two families in P via small Linear Code Equivalence instances, plus the external Luks 1980/1982 group-theoretic technique. The paper's contribution is an independent circuit implementation that exploits the given multiplication table to achieve depth 3; this step does not reduce to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation. The byproduct AC bound for central-radical groups is likewise an improvement on a prior external time bound rather than a re-derivation of the input. No equations or claims in the provided text exhibit the forbidden patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard facts from group theory and circuit complexity; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Groups are given by multiplication tables
    Stated in the abstract as the input model that enables the AC³ implementation.
  • standard math Luks' group-theoretic method for Graph Isomorphism applies to the relevant code-equivalence instances
    Invoked to obtain the parallel bound.

pith-pipeline@v0.9.0 · 5577 in / 1233 out tokens · 20293 ms · 2026-05-10T11:45:49.127353+00:00 · methodology

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Reference graph

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