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.

翻译：本文研究无阿贝尔正规子群（即无挠群）的最小忠实置换表示的计算复杂性。当群以置换群的商群形式给出时，我们构造了一个多项式时间算法来生成此类表示。进一步地，在置换群设定下，我们提出了一个$\textsf{NC}$过程来计算最小忠实置换度，以及一个随机化$\textsf{NC}$（$\textsf{RNC}$）算法来计算最小忠实置换表示。这改进了Das和Thakkar（STOC 2024, SIAM J. Comput. 2026）的工作，他们为置换群设定下的此类群建立了一个拉斯维加斯多项式时间算法来计算最小忠实置换度。