  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.
