Complexity of Constructing Minimal Faithful Permutation Representations for Fitting-free Groups

Dhara Thakkar; Michael Levet; Pranjal Srivastava

arxiv: 2501.16039 · v4 · submitted 2025-01-27 · 💻 cs.DS · cs.CC· math.GR

Complexity of Constructing Minimal Faithful Permutation Representations for Fitting-free Groups

Michael Levet , Pranjal Srivastava , Dhara Thakkar This is my paper

Pith reviewed 2026-05-23 05:14 UTC · model grok-4.3

classification 💻 cs.DS cs.CCmath.GR

keywords Fitting-free groupsminimal faithful permutation representationscomputational group theoryNC algorithmspermutation groupsquotient presentations

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The pith

Fitting-free groups given as quotients of permutation groups admit polynomial-time construction of minimal faithful permutation representations.

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

The paper establishes that for groups without nontrivial abelian normal subgroups, the task of building a smallest-degree faithful action on a set becomes tractable under standard input presentations. When the input is a quotient of a permutation group, a deterministic polynomial-time procedure outputs the representation. When the input is already a permutation group, the minimal degree is computable in NC while the representation itself is obtainable in RNC. These procedures improve earlier Las Vegas polynomial-time results for the same class by removing randomness and moving into the parallel setting.

Core claim

For Fitting-free groups presented as quotients of permutation groups, a polynomial-time algorithm constructs a minimal faithful permutation representation. For the same groups presented directly as permutation groups, the minimal faithful permutation degree is computable in NC and a minimal faithful permutation representation is computable in RNC.

What carries the argument

The minimal faithful permutation representation, whose degree is determined by the smallest faithful action arising from the socle of a Fitting-free group.

If this is right

The minimal faithful degree becomes an efficiently computable invariant for this class of groups.
Parallel algorithms become available for constructing smallest faithful actions in the permutation-group model.
Earlier Las Vegas polynomial-time methods are strengthened to deterministic NC for the degree and RNC for the representation.
Structural features of Fitting-free groups suffice to bypass the need for exhaustive search over possible actions.

Where Pith is reading between the lines

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

The same structural reduction may apply to computing other minimal actions once the socle is known.
Extending the approach to groups that do possess abelian normal subgroups would require separate handling of those factors.
The NC and RNC bounds suggest that the problem lies in the parallel complexity hierarchy in a way that could be useful for circuit-based group algorithms.

Load-bearing premise

The input groups have no nontrivial abelian normal subgroups and are supplied either as quotients of permutation groups or as permutation groups themselves.

What would settle it

A concrete Fitting-free group given as a permutation group on which the minimal faithful degree cannot be computed in polylogarithmic parallel time.

read the original abstract

In this paper, we investigate the complexity of computing minimal faithful permutation representations for groups without abelian normal subgroups (a.k.a. Fitting-free groups). When our groups are given as quotients of permutation groups, we exhibit a polynomial-time algorithm for constructing such representations. Furthermore, in the setting of permutation groups, we obtain an $\textsf{NC}$ procedure for computing the minimal faithful permutation degree, and a randomized $\textsf{NC}$ ($\textsf{RNC}$) algorithm for computing a minimal faithful permutation representation. This improves upon the work of Das and Thakkar (STOC 2024, SIAM J. Comput. 2026), who established a Las Vegas polynomial-time algorithm for computing the minimal faithful permutation degree for this class in the setting of permutation groups.

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.

Desk Editor's Note private letter to a colleague

NC and RNC algorithms for minimal faithful permutation representations of Fitting-free groups, improving on prior Las Vegas poly-time bound.

read the letter

The main takeaway is that this paper supplies an NC algorithm for the minimal faithful permutation degree and an RNC algorithm for the representation itself when the input is a permutation group, plus a deterministic polynomial-time construction when the group is given as a quotient of a permutation group. Both results apply only to Fitting-free groups. This moves the perm-group case from the Las Vegas polynomial-time bound in Das and Thakkar to deterministic parallel classes, which is the concrete advance.

Referee Report

0 major / 3 minor

Summary. The manuscript claims a polynomial-time algorithm to construct minimal faithful permutation representations for Fitting-free groups given as quotients of permutation groups. For groups given directly as permutation groups, it claims an NC algorithm to compute the minimal faithful permutation degree and an RNC algorithm to compute a minimal faithful permutation representation, improving on the prior Las Vegas polynomial-time bound of Das and Thakkar.

Significance. If correct, the results advance computational group theory by placing the construction of minimal faithful permutation representations for Fitting-free groups (which reduce to direct products of non-abelian simple groups) into standard parallel complexity classes. The NC and RNC bounds constitute a clear improvement over sequential Las Vegas polynomial time and could enable more efficient handling of large instances in this class.

minor comments (3)

[Abstract] Abstract: the high-level claim that the algorithms exist is clear, but a one-sentence pointer to the main technical reduction (e.g., to composition factors or known minimal-degree formulas for simple groups) would help readers assess the scope immediately.
The manuscript should explicitly state the precise input encoding (generators plus normal subgroup or quotient presentation) in the theorem statements rather than only in the introduction, to avoid ambiguity about the computational model.
A short comparison table or paragraph contrasting the new NC/RNC bounds with the prior Las Vegas polynomial-time result (including the role of randomization) would clarify the improvement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The report lists no major comments, so we have nothing to address point by point at this stage.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states algorithmic existence and complexity results (poly-time construction for quotient inputs; NC for minimal degree and RNC for the representation under permutation-group inputs) for Fitting-free groups. These rest on the standard computational models and the Fitting-free hypothesis permitting reduction to direct products of non-abelian simple groups, without any equations, fitted parameters, self-definitional quantities, or load-bearing self-citations that collapse the new claims to prior unverified results. The reference to Das and Thakkar is used only to contextualize the improvement and is not required to establish the stated NC/RNC bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions of computational group theory and parallel complexity; no free parameters or invented entities are introduced.

axioms (2)

domain assumption Groups are presented in one of the two standard computational models (permutation groups or quotients thereof).
The algorithms are stated only for these input representations.
standard math Standard definitions of NC and RNC complexity classes apply.
The paper invokes these classes to classify its algorithms.

pith-pipeline@v0.9.0 · 5666 in / 1244 out tokens · 46669 ms · 2026-05-23T05:14:12.473519+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2 and Proposition 3.1: NC computation of Soc(G) and μ(G) for Fitting-free permutation groups via semisimple series and centralizers.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Use of J-cost or recognition-cost reasoning is absent; only classical permutation-group algorithms (BLS87, KL90a) appear.

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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