Suppose that G is a group with less than 1,000,000 elements. Show that you can find no more than 20 elements that generate G.
As the minimal order of non-unit element in a group is 2, and as the minimal number of elements a group with 20 generators is $\displaystyle \left|\mathbb{Z}_2\times\ldots\times\mathbb{Z}_2\r ight|=2^{20}>1,000,000$ ,
with 20 direct factors there (why?), then we're done.
Tonio