  Suppose that G is a finite group and x in G has prime order p > 3. Then x is
contained in the solvable radical of G if (and only if) <x,x^g> is solvable for
all g in G. If G is an almost simple group and x in G has prime order p > 3
then this implies that there exists g in G such that <x,x^g> is not solvable.
In fact, this is also true when p=3 with very few exceptions, which are
described explicitly.
