Complexity of Minimal Faithful Permutation Degree for Fitting-free Groups

2025-01-27

AI Summary

This paper investigates the computational complexity of computing the minimal faithful permutation degree of a finite group—the smallest size of a set on which the group acts faithfully. For finite groups with no nontrivial abelian normal subgroups, we present the first deterministic polynomial-time algorithm under the quotient-group representation model. Under the standard permutation-group input model, we improve the best-known upper bound from Las Vegas polynomial time (Das–Thakkar, STOC 2024) to NC—yielding the first deterministic parallel algorithm for this problem. Our approach integrates structural group-theoretic analysis, efficient computation of quotient-group homomorphisms, and advanced permutation-group algorithms, thereby overcoming prior reliance on randomness. The result unifies treatment across a broader class of groups and establishes a new complexity benchmark for efficient computation of group actions.

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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 $ extsf{P}$. Furthermore, in the setting of permutation groups, we obtain an upper bound of $ extsf{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.

Problem

• Fitting-free groups
• faithful transformation
• minimum permutation degree

Innovation

• Fitting-free groups
• Efficient computation
• Optimized transformation