有限群的博弈导体：来自结构化收益探测的行列式挠率
Game Conductors of Finite Groups: Determinantal Torsion from Structured Payoff Probes
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中文总结 AI 辅助
研究有限群\(G\)与结构化收益探测\(\phi\)相关的收益差格及导体，通过精确计算得出CA - 群交换导体公式，利用伽罗瓦轨道迹特征探测补充结果，经详尽计算和分析支持对压缩中心化子型关联矩阵\(\BG\)的史密斯挠率进行分类。
中文摘要 AI 辅助
我们为有限群\(G\)和结构化收益探测\(\phi\)附上一个整数“收益差格”\(M_\phi(G)\)及其“导体”\(C_\phi(G)\)，即\(M_\phi(G)\)在模\(p\)下失去秩的素数。主要结果是精确计算：对于任何CA - 群，交换导体是\(\rad(b - 1)\)，其中\(b\)是极大阿贝尔子群的数量。交换史密斯谱是同构类的不变量且遵循精确的直积定律。通过伽罗瓦轨道迹特征探测得到补充结果，如指数为\(2\)的子群迫使\(2\in\Cchar(G)\)等。经认证的详尽计算和变形族分析支持对压缩中心化子型关联矩阵\(\BG\)的史密斯挠率进行分类的总体计划。
英文摘要
We attach to a finite group $G$ and a structured payoff probe $ϕ$ an integer \emph{payoff-difference lattice} $M_ϕ(G)$ and its \emph{conductor} $C_ϕ(G)$: the primes at which $M_ϕ(G)$ loses rank modulo $p$. Our main result is an exact computation: for any CA-group the commuting conductor is rad$(b-1)$, where $b$ is the number of maximal abelian subgroups. In particular, conductor primes need not divide $|G|$: the prime $3$ occurs for a $2$-group of order $64$ with $b=7$. The commuting Smith spectrum is an invariant of the isoclinism class and obeys an exact direct-product law, giving ${\rm C_{comm}}(G\times H) = {\rm C_{comm}}(G) \cup {\rm C_{comm}}(H)$ unconditionally. A Galois-orbit-trace character probe reads a complementary layer: an index-$2$ subgroup forces $2\in {\rm C_{char}}(G)$ while no odd prime is forced, and ${\rm C_{comm}}(D_{2q}) = \{q\}$, ${\rm C_{char}}(D_{2q}) = \{2\}$ for all odd primes $q$. Certified exhaustive computation ($|G|\le128$ commuting, $|G|\le64$ character) and a deformation-family analysis support the general program: classify the Smith torsion of the compressed centralizer-type incidence matrix $B_G$.
