论文标题
缺陷1和Zassenhaus猜想的单位
Units in Blocks of Defect 1 and the Zassenhaus Conjecture
论文作者
论文摘要
在Caicedo和第二作者的先前工作的基础上，我们开发了一种方法，该方法在$ \ Mathbb {z} _p g $ of Defect 1的块中存在有限订单的存在。 $ P $与$ \ pm g $的元素共轭。这是Zassenhaus猜想的特殊情况。我们还以$ \ mathbb {z} \ mathrm {psl}（2，q）$的某些$ q $的有限订单单位证明了一些新结果，并为某些$ q $构建了$ 15 $ $ 15 $的单位$ v（\ mathbb {z} _ {z} _ {（3,3,5）} \ mathrm {$ 3 $ {$ 3 $ 3，2,16）对Zassenhaus猜想的反例，提出了希望我们的方法可能导致简单组之间的全球反例。
Building on previous work by Caicedo and the second author, we develop a method that decides the existence of units of finite order in blocks of $\mathbb{Z}_p G$ of defect 1. This allows us to prove that if $p$ is a prime and $G$ is a finite group whose Sylow $p$-subgroup has order $p$, then any unit $u\in \mathbb{Z} G$ of order $p$ is conjugate to an element of $\pm G$. This is a special case of the Zassenhaus conjecture. We also prove some new results on units of finite order in $\mathbb{Z} \mathrm{PSL}(2,q)$ for certain $q$, and construct a unit of order $15$ in $V(\mathbb{Z}_{(3,5)}\mathrm{PSL}(2,16))$ which is a $3$- and $5$-local counterexample to the Zassenhaus conjecture, raising the hope that our methods may lead to a global counterexample amongst simple groups.
