We consider finite-sample inference for a single regression coefficient in the fixed-design linear model $Y = Zβ+ bX + \varepsilon$, where $\varepsilon\in\mathbb{R}^n$ may exhibit complex dependence or heterogeneity. We develop a group permutation framework, yielding a unified and analyzable randomization structure for linear-model testing. Under exchangeable errors, we place permutation-augmented regression tests within this group-theoretic setting and show that a grouped version of PALMRT controls Type I error at level at most $2α$ for any permutation group; moreover, we provide an worst-case construction demonstrating that the factor $2$ is sharp and cannot be improved without additional assumptions. Second, we relate the Type II error to a design-dependent geometric separation. We formulate it as a combinatorial optimization problem over permutation groups and bound it under additional mild sub-Gaussian assumptions. For the Type II error upper bound control, we propose a constructive algorithm for the permutation strategy that is better (at least no worse) than the i.i.d. permutation, with simulations empirically indicating substantial power gains, especially under heavy-tailed designs. Finally, we extend group-based CPT and PALMRT beyond exchangeability by connecting rank-based randomization arguments to conformal inference. The resulting weighted group tests satisfy finite-sample Type I error bounds that degrade gracefully with a weighted average of total variation distances between $\varepsilon$ and its group-permuted versions, recovering exact validity when these discrepancies vanish and yielding quantitative robustness otherwise. Taken together, the group-permutation viewpoint provides a principled bridge from exact randomization validity to design-adaptive power and quantitative robustness under approximate symmetries.

翻译：我们考虑固定设计线性模型$Y = Zβ+ bX + \varepsilon$中单个回归系数的有限样本推断，其中$\varepsilon\in\mathbb{R}^n$可能呈现复杂依赖性或异质性。我们建立了组置换框架，为线性模型检验提供了统一且可分析的随机化结构。在误差可交换条件下，我们将置换增强回归检验置于该群论框架中，证明分组版PALMRT对任意置换群均能以最多$2α$的水平控制第一类错误；进一步地，我们通过最坏情况构造证明因子$2$具有尖锐性且无法在无额外假设条件下改进。其次，我们将第二类误差与设计相关的几何分离建立联系，将其表述为置换群上的组合优化问题，并在附加温和次高斯假设下给出其界。为控制第二类误差上界，我们提出一种构造性置换策略算法，其性能优于（至少不差于）独立同分布置换，仿真实验表明该方法能带来显著功效提升，尤其在重尾设计条件下。最后，我们通过将基于秩的随机化论证与保形推断相结合，将基于组的CPT和PALMRT扩展至非可交换情形。所得加权组检验满足有限样本第一类误差界，该误差界随$\varepsilon$与其组置换版本间总变差距离加权平均值平缓退化——当这些差异消失时恢复精确有效性，否则提供定量鲁棒性。综上所述，组置换视角为近似对称性下的精确随机化有效性、设计自适应功效与定量鲁棒性之间建立了原则性桥梁。