论文标题
Shintani血统，简单的群体和传播
Shintani descent, simple groups and spread
论文作者
论文摘要
$ g $，书面$ s（g）$的传播是最大的$ k $，因此对于任何非平凡元素$ x_1，\ dots，x_k \ in g $ in G $中的$ y \ in g $ in g $ in g $ y \ in g $ y \ in g $ y \ y \ langle x_i x_i，y \ rangle $ for All $ i $ $ $。 Burness，Guralnick和Harper最近将有限的组$ g $分类为$ s（g）> 0 $，这涉及减少几乎简单的群体。在本文中，我们证明了一个渐近结果，该结果准确地确定了$ s（g_n）\ to \ infty $的几乎简单组（g_n）$的序列。我们采用概率和几何思想，但关键工具是Shintani Descent，这是代数群体理论的一种技术，该技术在几乎简单的群体的共轭类别之间提供了两者的培养物，即Shintani Map。我们提供了Shintani Descent的一般版本的独立陈述，我们证明Shintani地图保留了有关最大杂物的信息。这适合进一步的申请。的确，我们还使用它来研究$μ（g）$，这是$ g $的最小数量。我们表明，如果$ g $几乎很简单，则$μ（g）\ leqslant 3 $当$ g $具有交替或零星的socle时，但是通常，与$ g $很简单时，$μ（g）$可以任意大。
The spread of a group $G$, written $s(G)$, is the largest $k$ such that for any nontrivial elements $x_1, \dots, x_k \in G$ there exists $y \in G$ such that $G = \langle x_i, y \rangle$ for all $i$. Burness, Guralnick and Harper recently classified the finite groups $G$ such that $s(G) > 0$, which involved a reduction to almost simple groups. In this paper, we prove an asymptotic result that determines exactly when $s(G_n) \to \infty$ for a sequence of almost simple groups $(G_n)$. We apply probabilistic and geometric ideas, but the key tool is Shintani descent, a technique from the theory of algebraic groups that provides a bijection, the Shintani map, between conjugacy classes of almost simple groups. We provide a self-contained presentation of a general version of Shintani descent, and we prove that the Shintani map preserves information about maximal overgroups. This is suited to further applications. Indeed, we also use it to study $μ(G)$, the minimal number of maximal overgroups of an element of $G$. We show that if $G$ is almost simple, then $μ(G) \leqslant 3$ when $G$ has an alternating or sporadic socle, but in general, unlike when $G$ is simple, $μ(G)$ can be arbitrarily large.
