论文标题
正常子集和毁灭的产物
Products of normal subsets and derangements
论文作者
论文摘要
近年来，在研究有限群体尤其是有限简单组的子集的产品的研究中取得了重大进展。在本文中，我们考虑哪些有限的简单群体$ g $具有每个$ε> 0 $的财产，那里存在$ n> 0 $，这样，如果$ | g | \ ge n $和$ s，t $是$ g $的普通子集，每个元素至少为$ε| $元素，那么$ g $的每个非平凡元素都是$ s $的元素和$ t $的元素的产物。 我们表明，这对于有限的谎言类型等级的有限简单组具有很强的意义，而对于$ {\ mathrm {psl}} _ n（q）$ $ q $的交替组或组的交替组或组不满意，而$ q $是固定的，而$ n $倾向于无限。 我们的第二个主要结果是，足够大的有限简单组的及当置换置换表示中的任何元素都是两个扰动的产物。
In recent years there has been significant progress in the study of products of subsets of finite groups and of finite simple groups in particular. In this paper we consider which families of finite simple groups $G$ have the property that for each $ε> 0$ there exists $N > 0$ such that, if $|G| \ge N$ and $S, T$ are normal subsets of $G$ with at least $ε|G|$ elements each, then every non-trivial element of $G$ is the product of an element of $S$ and an element of $T$. We show that this holds in a strong sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form ${\mathrm{PSL}}_n(q)$ where $q$ is fixed and $n$ tends to infinity. Our second main result is that any element in a transitive permutation representation of a sufficiently large finite simple group is a product of two derangements.
