The spread of a finite group
The spread of a finite group
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DOI：
10.4007/annals.2021.193.2.5
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发表时间：
2020-06
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通讯作者：
Timothy C. Burness;R. Guralnick;Scott Harper
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文献类型：
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作者：
Timothy C. Burness;R. Guralnick;Scott Harper
一个群$G$被称为$\frac{3}{2}$-生成的，如果每个非平凡元素都属于一个生成对。很容易看出，如果$G$具有这个性质，那么$G$的每一个真商都是循环的。在本文中，我们证明了有限群的匡威命题成立，从而解决了Breuer，Guralnick和Kantor在2008年提出的一个猜想。事实上，我们证明了一个强得多的结果，解决了Brenner和Wiegold在1975年提出的一个问题。即，如果$G$是有限群并且$G$的每个真商都是循环的，则对于G$中的任何一对非平凡元素$x_1，x_2\，G $中存在$y \，使得$G = \langle x_1，y \rangle = \langle x_2，y \rangle$。换句话说，$s（G）\geqslant 2$，其中$s（G）$是$G$的价差。此外，如果u（G）$表示G$的更严格的一致扩展，则我们可以完全地刻画u（G）= 0$和u（G）=1$的有限群G$。为了证明这些结果，我们首先建立了几乎单群的约化。对于单群，Guralnick和Kantor在2000年用概率方法证明了这个结果，从那时起几乎单群就成为了几篇论文的主题。通过结合我们的约化定理和这一早期的工作，它仍然是处理组的socles是例外群的李型，这是我们在本文中治疗的情况。
A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if $G$ is a finite group and every proper quotient of $G$ is cyclic, then for any pair of nontrivial elements $x_1,x_2 \in G$, there exists $y \in G$ such that $G = \langle x_1, y \rangle = \langle x_2, y \rangle$. In other words, $s(G) \geqslant 2$, where $s(G)$ is the spread of $G$. Moreover, if $u(G)$ denotes the more restrictive uniform spread of $G$, then we can completely characterise the finite groups $G$ with $u(G) = 0$ and $u(G)=1$. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper.