论文标题
有限群体的传播
The spread of a finite group
论文作者
论文摘要
如果每个非平凡元素都属于生成对，则据说$ g $是$ \ frac {3} {2} $ - 生成的。很容易看出，如果$ g $具有此属性，那么$ g $的每一个适当的商都是循环的。在本文中，我们证明了相反的群体是正确的，该群体解决了2008年的Breuer，Guralnick和Kantor的猜想。实际上，我们证明了一个更强的结果，这可以解决Brenner和Wiegold在1975年提出的问题。 $ x_1，x_2 \在g $中，存在$ y \ in g $，使得$ 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 $的差价。此外，如果$ u（g）$表示$ g $的更限制的统一点差，那么我们可以完全表征有限的组$ g $，$ u（g）= 0 $和$ u（g）= 1 $。为了证明这些结果，我们首先将减少到几乎简单的群体中。对于简单的组，Guralnick和Kantor在2000年使用概率方法证明了结果，从那时起，几乎简单的组就成为了几篇论文的主题。通过将我们的还原定理和早期工作结合起来，仍然要处理SOCLE是谎言类型的杰出群体的群体，而我们在本文中就是这种情况。
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.
