一般线性群在非定义特征标下的内平凡模
Endotrivial Modules for the General Linear Group in a Nondefining Characteristic
摘要：
设$G$是一个有限群，使得$\operatorname{SL}(n,q)\subseteq G \subseteq \operatorname{GL}(n,q)$，且$Z$是$G$的中心子群.设$T(G/Z)$是内平凡$k(G/Z)$ -模的等价类的交换群，其中$k$是特征~ $p$不除$q$的代数闭域.证明了$T(G/Z)$的挠自由秩至多为1，并在$G$的Sylow $p$ -子群为交换非平凡的情形下，确定了$T(G/Z)$. $T(G/Z)$的挠子群的证明使用了$\operatorname{GL}(n,q)$的Young模理论和Balmer提出的一种新的方法来计算Endovrivial模群中的限制核。
英文摘要：
Suppose that $G$ is a finite group such that $\operatorname{SL}(n,q)\subseteq G \subseteq \operatorname{GL}(n,q)$, and that $Z$ is a central subgroup of $G$. Let $T(G/Z)$ be the abelian group of equivalence classes of endotrivial $k(G/Z)$-modules, where $k$ is an algebraically closed field of characteristic~$p$ not dividing $q$. We show that the torsion free rank of $T(G/Z)$ is at most one, and we determine $T(G/Z)$ in the case that the Sylow $p$-subgroup of $G$ is abelian and nontrivial. The proofs for the torsion subgroup of $T(G/Z)$ use the theory of Young modules for $\operatorname{GL}(n,q)$ and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
