Exhibit all of the -module homomorphisms from to .
By this previous exercise, we have . We claim that . To see this, suppose first that . In particular, we have mod 21. Then , and so . Now mod 7, and hence mod 7. So . Conversely, note that . Thus , and we have .
So there are precisely three -module homomorphisms . As group homomorphisms, each of these is uniquely determined by the image of . These homomorphisms are given by , , and .