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Complexity of Minimal Faithful Permutation Degree for Fitting-free Groups

Michael Levet, Pranjal Srivastava, Dhara Thakkar

TL;DR

The paper resolves the complexity of computing the minimal faithful permutation degree $\mu(G)$ for FF groups by showing polynomial-time solvability when inputs are quotients of permutation groups and $\mathsf{NC}$-computability for permutation-group inputs. It develops an $\mathsf{NC}$-capable method to compute $\mathrm{Soc}(G)$ for FF groups, and provides lifting procedures to extend automorphisms through key simple groups, enabling constructive recognition steps in parallel. A central contribution is reducing $\mu(G)$ to a sum of values $\mu(G,N_i)$ over minimal normal subgroups, with each $\mu(G,N_i)$ computed via the conjugation action on simple factors and known table-driven results (Cannon–Holt–Unger). The work culminates in a circuit-complexity bound that places Min-Per-Deg in a restricted $\mathsf{AC}^{0}$-augmented framework and highlights a barrier to $\mathsf{GI}$-hardness reductions, while leaving open the precise $\mathsf{NC}$-membership for the quotients model. Overall, the results advance parallelizable approaches for permutation-group quotients and demarcate the limits of current NC techniques in this domain.

Abstract

In this paper, we investigate the complexity of computing the minimal faithful permutation degree for groups without abelian normal subgroups. When our groups are given as quotients of permutation groups, we establish that this problem is in $\textsf{P}$. Furthermore, in the setting of permutation groups, we obtain an upper bound of $\textsf{NC}$ for this problem. This improves upon the work of Das and Thakkar (STOC 2024), who established a Las Vegas polynomial-time algorithm for this class in the setting of permutation groups.

Complexity of Minimal Faithful Permutation Degree for Fitting-free Groups

TL;DR

The paper resolves the complexity of computing the minimal faithful permutation degree for FF groups by showing polynomial-time solvability when inputs are quotients of permutation groups and -computability for permutation-group inputs. It develops an -capable method to compute for FF groups, and provides lifting procedures to extend automorphisms through key simple groups, enabling constructive recognition steps in parallel. A central contribution is reducing to a sum of values over minimal normal subgroups, with each computed via the conjugation action on simple factors and known table-driven results (Cannon–Holt–Unger). The work culminates in a circuit-complexity bound that places Min-Per-Deg in a restricted -augmented framework and highlights a barrier to -hardness reductions, while leaving open the precise -membership for the quotients model. Overall, the results advance parallelizable approaches for permutation-group quotients and demarcate the limits of current NC techniques in this domain.

Abstract

In this paper, we investigate the complexity of computing the minimal faithful permutation degree for groups without abelian normal subgroups. When our groups are given as quotients of permutation groups, we establish that this problem is in . Furthermore, in the setting of permutation groups, we obtain an upper bound of for this problem. This improves upon the work of Das and Thakkar (STOC 2024), who established a Las Vegas polynomial-time algorithm for this class in the setting of permutation groups.

Paper Structure

This paper contains 23 sections, 33 theorems, 22 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Groups
  4. Computational Complexity
  5. Permutation Group Algorithms
  6. Background on Simple Groups
  7. Constructive Recognition Algorithm for Classical Simple Groups
  8. The Minimal Faithful Permutation degree of Simple and Almost simple groups
  9. Background on Representation Theory
  10. Computing Socle of Fitting-free Group
  11. Lifting of Automorphisms
  12. Lifting Automorphisms of $\text{PSL}(d,q)$ to Automorphisms of $\text{SL}(d,q)$
  13. Lifting Automorphisms of $\mathrm{P}\Omega^{+}(2d,3)$ to Automorphisms of $\Omega^+(2d,3)$
  14. The minimal faithful permutation degree of Fitting-free groups
  15. .
  16. ...and 8 more sections

Key Result

Proposition 1.1

Let $G$ be a finite group, and let $\mathcal{G}$ be the collection of all subgroups of $G$. We have that: where $\text{Core}_{G}(H) = \bigcap_{g \in G} gHg^{-1}$ and $|\mathcal{H}| \leq \log |G|$.The fact that we can require any witnessing collection of subgroups $\mathcal{H}$ to satisfy $|\mathcal{H}| \leq \log |G|$ was carefully analyzed by DasThakkarMPD.

Theorems & Definitions (59)

  • Proposition 1.1: JohnsonMPD
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2: BabaiLuksSeress
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 49 more