发表机构
Texas State University（德克萨斯州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究奇阶非交换线性群的最大轨道大小与交换商之比，证明一般有 $|G:G'|\le M/p$，例外情形为 $3M/7$，并给出更精细的界及最优性。
AI 中文摘要
设 $G$ 为有限非交换奇阶群，$p$ 为 $|G|$ 的最小素因子，$V$ 为有限忠实完全可约 $G$-模，可能具有混合特征。假设 $M$ 是 $G$ 在 $V$ 上作用的最大轨道大小。已知 $|G:G'|\le M$，且当 $|G|$ 为奇数时，等号仅对交换群成立。我们证明 $|G:G'|\le M/p$，除非 $p=3$ 且 $G$ 是交换群与同构于 $\Gamma(2^3)$ 或 $\Gamma(2^3)\times C_3$ 的群的直积，并以特定方式作用于 $V$，此时 $|G:G'|=3M/7$。特别地，对每个非交换奇阶群，$|G:G'|\le 3M/7$。若 $|V|$ 为奇数，则 $|G:G'|=M/p$ 当且仅当 $|G'|=p$；更精确地，幂零群此时具有正则轨道，而非幂零群满足 $|G:G'|\le 3M/13$，且若 $p\ge5$，甚至 $|G:G'|<M/(2p)$。界 $3M/13$ 是最优的。
英文摘要
Let $G$ be a finite nonabelian group of odd order, let $p$ be the smallest prime divisor of $|G|$, and let $V$ be a finite faithful completely reducible $G$-module, possibly of mixed characteristic. Suppose that $M$ is the largest orbit size in the action of $G$ on $V$. It is known that $|G:G'|\le M$, and that equality holds only for abelian groups if $|G|$ is odd. We prove that $|G:G'|\le M/p$, unless $p=3$ and $G$ is the direct product of an abelian group and a group isomorphic to $Γ(2^3)$ or to $Γ(2^3)\times C_3$ acting on $V$ in a specific way, in which case $|G:G'|=3M/7$. In particular, $|G:G'|\le 3M/7$ for every nonabelian group of odd order. If $|V|$ is odd, then $|G:G'|=M/p$ holds if and only if $|G'|=p$; more precisely, nilpotent groups then have regular orbits, whereas non-nilpotent groups satisfy $|G:G'|\le 3M/13$, and even $|G:G'|<M/(2p)$ if $p\ge5$. The bound $3M/13$ is best possible.
