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

TL;DR

This paper studies the computational complexity of finding minimal faithful permutation representations for Fitting-free groups, providing efficient algorithms and improving previous results.

Contribution

It introduces polynomial-time and parallel algorithms for constructing minimal faithful permutation representations of Fitting-free groups, advancing computational group theory.

Findings

Polynomial-time algorithm for constructing representations from quotient groups.

NC procedure for computing minimal faithful permutation degree.

RNC algorithm for minimal faithful permutation representation.

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.