  We prove that there exists a constant $k$ with the property: if $\calC$ is a
conjugacy class of a finite group $G$ such that every $k$ elements of $\calC$\
generate a solvable subgroup then $\calC$ generates a solvable subgroup. In
particular, using the Classification of Finite Simple Groups, we show that we
can take $k=4$. We also present proofs that do not use the Classification
theorem. The most direct proof gives a value of $k=10$. By lengthening one of
our arguments slightly, we obtain a value of $k=7$.
