In 1968, John Thompson proved that a finite group G is solvable if and only
if every 2-generator subgroup of G is solvable. In this paper, we prove that
solvability of a finite group G is guaranteed by a seemingly weaker condition:
G is solvable if, for all conjugacy classes C and D of G consisting of elements
of prime power order, there exist x in C and y in D with x and y generating a
solvable group. We also prove the following property of finite nonabelian
simple groups, which is the key tool for our proof of the solvability
criterion: if G is a finite nonabelian simple group, then there exist two prime
divisors a and b of |G| such that, for all elements x, y in G with |x|=a and
|y|=b, the subgroup generated by x and y is not solvable. Further, using a
recent result of Guralnick and Malle, we obtain a similar membership criterion
for any family of finite groups closed under forming subgroups, quotients and
extensions.