The spread of a finite group

A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs a generating pair. It easy see that has this property, then proper quotient of cyclic. In paper we prove the converse true for finite groups, which settles conjecture Breuer, Guralnick and Kantor from 2008. fact, much stronger result, solves problem posed by Brenner Wiegold in 1975. Namely, cyclic, any pair elements $x_1,x_2 \in G$, there exists $y\in G$ such $G = \langle x_1y\rangle x_2,y\rangle$. other words, $s(G) \ge 2$, where $s(G)$ spread $G$. Moreover, $u(G)$ denotes more restrictive uniform $G$, can completely characterise groups with $u(G) 0$ $u(G)=1$. To these results, first establish reduction almost simple groups. For result was proved 2000 using probabilistic methods, since have been subject several papers. By combining our theorem earlier work, it remains handle socle an exceptional Lie type, case treat paper.