Hello, I have an abstract algebra question.

I need your help.

Thank you so much.

*4. (8 points) Let G be a finite abelian group of order n (written additively). For each a E G, we then have that na = 0. In particular, if i E Z, then the map fli : G - G, am>ia depends only on the class [i] of i in Zin. (a) Show that if i E Z, then fa is a group homomorphism, and an automorphism of G in the case where god(i, n) = 1. (b) By (a), we have a map f : Zx -> Aut(G), -> fli, where Z, := { [i] E Zn | god(i, n) = 1}. Recall from Math 212 that Zx can be equipped with a group structure by setting [j] = [ij] for all integers i and j coprime to n. Show that if we view Z, as a group in this way, then f is a group homomorphism.(c) Show that if G is cyclic, then the group homomorphism f from part (b) is an isomorphism. In particular, we have |Aut(G) | = p(n) in this case (4 being Euler's totient function)