循环洗牌群：通用双传递性与完全分类
Cyclic Shuffle Groups: Universal Two-Transitivity and Complete Classification
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中文总结 AI 辅助
该论文研究由k堆完美洗牌与循环堆置换生成的群H_{k,n}，证明其在n非k幂时为2-传递，完成H_{k,n}及含C_k的堆群P对应的Sh(P,n)的完全分类，解决相关群论猜想。
中文摘要 AI 辅助
设k≥3，n≥1，令H_{k,n}=Sh(C_k,n)为标准k堆完美洗牌与kn张牌堆上的循环堆置换生成的群。我们证明，当n不是k的幂时，H_{k,n}是2-传递的。剩余交换子给出支撑在两个堆标号上的平移，强连通数字有向图将这些平移传播至整个牌堆，单独的论证解决了对跖支撑情形。随后，我们将该结果与本原群的不动点比率界及显式边界计算相结合，确定所有k和n对应的H_{k,n}：若n=k^f，则H_{k,n}≅C_k≀C_{f+1}；若k=4且n=2·4^j，则H_{4,n}≅AGL(2j+3,2)；其余所有情形下，H_{k,n}依生成元的奇偶性为Alt(kn)或Sym(kn)。这证明了Amarra、Morgan与Praeger的猜想1.10，以及Xia、Zhang、Zhang与Zhu的猜想5.1。更一般地，我们对每个包含C_k的堆群P分类Sh(P,n)，并得到其猜想5.2的奇数k部分。
英文摘要
Let \(k\geq 3\), \(n\geq 1\), and let \(H_{k,n}=\Sh(C_k,n)\) be the group generated by the standard \(k\) pile perfect shuffle and cyclic pile permutation on a deck of \(kn\) cards. We prove that \(H_{k,n}\) is \(2\)-transitive whenever \(n\) is not a power of \(k\). Residual commutators give translations supported on two pile labels, and a strongly connected digit digraph propagates these translations throughout the deck, a separate argument resolves the antipodal support case. We then combine this result with fixed point ratio bounds for primitive groups and explicit boundary calculations to determine \(H_{k,n}\) for all \(k\) and \(n\). If \(n=k^f\), then \(H_{k,n}\cong C_k\wr C_{f+1}\). If \(k=4\) and \(n=2\cdot4^j\), then \(H_{4,n}\cong\AGL(2j+3,2)\). In every other case, \(H_{k,n}\) is \(\Alt(kn)\) or \(\Sym(kn)\), according to the parity of its generators. This proves Conjecture~1.10 of Amarra, Morgan and Praeger and Conjecture~5.1 of Xia, Zhang, Zhang and Zhu. More generally, we classify \(\Sh(P,n)\) for every pile group \(P\) containing \(C_k\), and obtain the odd \(k\) part of their Conjecture~5.2.
