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 with socle an exceptional group of Lie type, and this is the case we treat in this paper.

如果每个非平凡元素都属于生成对，则组$ G $被称为$ \ frac {3} {2} $生成。很容易看出，如果$ G $具有此属性，则$ G $的每个适当商都是循环的。在本文中，我们证明了有限群的逆是正确的，这解决了布劳尔，古拉尼克和康托尔从2008年以来的一个猜想。实际上，我们证明了更强的结果，解决了布伦纳和维戈德在1975年提出的问题。 ，如果$ G $是一个有限组，并且$ G $的每个适当商都是循环的，则对于G $中的任意一对非平凡元素$ x_1，x_2 \，在$$中都存在$ 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 $的限制更为严格的均匀利差，那么我们就可以用$ u（G）= 0 $和$ u（G）= 1 $来完全刻画有限群$ G $的特征。为了证明这些结果，我们首先确定了约简组的约简。对于简单的小组，Guralnick和Kantor在2000年使用概率方法证明了这一结果，从那时起，几乎简单的小组成为了几篇论文的主题。通过结合我们的约简定理和此早期工作，仍然可以处理具有唯一Lie型群的单数群，这就是我们在本文中要处理的情况。