13 pages. Comments welcome! v3 improves Lemma 6.7, following a suggestion from François Le Maître. v2 made minor changes to the exposition, some of which were suggested by an anonymous referee
A group is $\frac32$-generated if every non-trivial element is part of a generating pair. In 2019, Donoven and Harper showed that many Thompson groups are $\frac32$-generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\frac32$-generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups of the Houghton group $\text{FSym}(\mathbb{Z})\rtimes\mathbb{Z}$. We then show that the first two groups in our family are $\frac32$-generated, and investigate the related notion of spread for these groups. We are able to show that they have finite spread which is greater than 2. These are therefore the first infinite groups to be shown to have finite positive spread, and the first to be shown to have spread greater than 2 (other than $\mathbb{Z}$ and the Tarski monsters, which have infinite spread).
每个有限单群都可以由两个元素生成，在2000年，Guralnick和Kantor通过证明有限单群中的每个非平凡元素都属于一个生成对，解决了Steinberg在1962年提出的一个问题。具有此属性的组称为$\frac{3}{2}$生成的组。哪些有限群是$\frac{3}{2}$生成的？$\frac{3}{2}$生成群的每一个真商都是循环的，在2008年，Breuer，Guralnick和Kantor提出了一个惊人的猜想，这个条件单独提供了一个完整的有限群的特征。这个猜想最近被简化到几乎单群上，Piccard（1939）和Woldar（1994）的结果表明，这个猜想对于基元为交错群或散在群的几乎单群是成立的。因此，现在的中心焦点是几乎单群的李型。在本专著中，我们在Burness and Guest（2013）和作者（2017）的早期工作的基础上，对几乎简单的典型群证明了这个猜想的一个强版本。更精确地说，我们证明了每个相关的几乎单典型群至少有2个一致展度，除非它同构于6度的对称群。我们还证明了这些群体的均匀传播趋于无穷大，如果基础领域的大小趋于无穷大。为了证明这些结果，我们是由Guralnick和Kantor介绍的概率方法。这需要一个详细的分析自同构，不动点比率和子群结构几乎简单的典型群，所以上半年这本专着致力于这些一般性的议题。特别是，我们给出了一个一般性的阐述的有用的技术Shinta…
On the non-existence of certain transitive actions of solvable groups on symplectic spreads
我们证明了下面的结果。设q是奇素数的幂，Sp（2 m，q）表示F_q上2 m次的辛群。如果q=1 mod 4，则Sp（2m，q）的可解子群在定义于2m维向量空间上的完全辛展延上不传递，除非m=1且q=5。24阶可解群Sp（2，3）在F_5上的完全辛展布上传递作用。相比之下，当q=3 mod 4时，存在一个2m（q^m+1）阶的亚循环群传递地作用于完全辛展布上。
