Exercise 4.1.28

For prime, let be a functor sending a a group to its set of elements having order or . For a group homomorphism , simply let be the restriction of to – this is well defined since is prime, and thus necessarily sends an element of order to an element of order or .

Theorem: is representable. In particular, is isomorophic to .

Proof: Define a natural transformation as follows. For a group , let send an element of order or to the unique homomorphism such that . In other words, is the map . We now check naturality. Suppose is a homomorphism. We must have . Recall that is just left-composition by . Take a in . On the right side, we have , then left-composing with yields . On the left, we have . Then, . So both sides yield the same map , verifying that is natural. So .