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Invariable generation of finite classical groups
Journal of Algebra ( IF 0.9 ) Pub Date : 2021-06-29 , DOI: 10.1016/j.jalgebra.2021.06.020
Eilidh McKemmie

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements invariably generate Sn is bounded away from zero by an absolute constant for all n. Subsequently, Eberhard, Ford and Green have shown that the probability that three randomly selected elements invariably generate Sn tends to zero as n. In this paper, we prove an analogous result for the finite classical groups. More precisely, let Gr(q) be a finite classical group of rank r over Fq. We show that for q large enough, the probability that four randomly selected elements invariably generate Gr(q) is bounded away from zero by an absolute constant for all r, and for three elements the probability tends to zero as q and r. We use the fact that most elements in Gr(q) are separable and the well-known correspondence between classes of maximal tori containing separable elements in classical groups and conjugacy classes in their Weyl groups.



中文翻译:

有限经典群的不变生成

一个群的子集总是会生成这个群,即使我们用它们的任何一个共轭替换元素。在 2016 年的一篇论文中,Pemantle、Peres 和 Rivin 表明,四个随机选择的元素总是产生的概率n对于所有n,都被一个绝对常数限制在远离零的地方。随后,Eberhard、Ford 和 Green 证明了三个随机选择的元素总是产生的概率n 趋于零 n. 在本文中,我们证明了有限经典群的类似结果。更准确地说,让Gr(q)是秩的有限经典组řFq. 我们表明,对于足够大的q,四个随机选择的元素总是生成的概率Gr(q)对于所有r,由一个绝对常数限制远离零,并且对于三个元素,概率趋于零,如下所示qr. 我们使用的事实是,大多数元素在Gr(q) 是可分离的,并且是经典群中包含可分离元素的最大环面类与它们的外尔群中的共轭类之间的众所周知的对应关系。

更新日期:2021-06-29
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