论文标题
有限和无限组的传播
The spread of finite and infinite groups
论文作者
论文摘要
众所周知，每个有限的简单组都有生成对。此外，Guralnick和Kantor证明，每个有限的简单组具有更强的属性，称为$ \ frac {3} {2} $ - 生成，每个非平凡元素都包含在生成对中。最近，该结果已被概括为三个不同的方向，这构成了本调查文章的基础。首先，我们查看一些更强的形式的$ \ frac {3} {2} $ - 有限简单组满足的生成，这些形式是用涂抹和统一的统治来描述的。接下来，我们讨论有限$ \ frac {3} {2} $生成的组的最新分类。最后，我们将注意力转移到无限的群体上，重点是最近发现的发现，汤普森的简单群体也是$ \ frac {3} {2} $ - 生成的，就像他们的许多概括一样。在整篇文章中，我们在该领域提出了开放的问题，我们强调了与小组理论的其他领域的联系。
It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as $\frac{3}{2}$-generation, that every nontrivial element is contained in a generating pair. Much more recently, this result has been generalised in three different directions, which form the basis of this survey article. First, we look at some stronger forms of $\frac{3}{2}$-generation that the finite simple groups satisfy, which are described in terms of spread and uniform domination. Next, we discuss the recent classification of the finite $\frac{3}{2}$-generated groups. Finally, we turn our attention to infinite groups, focusing on the recent discovery that the finitely presented simple groups of Thompson are also $\frac{3}{2}$-generated, as are many of their generalisations. Throughout the article we pose open questions in this area, and we highlight connections with other areas of group theory.
