Problem 4

Let $G$ be a cyclic group of order $n$ with generator $g$.

1. Suppose the maps $f_{1}:G\to G$ and $f_{2}:G\to G$ are homomorphisms. Prove that $f_{1}=f_{2}$ if and only if $f_{1}(g)=f_{2}(g)$.

2. Let the map $f:G\to G$ be a homomorphism. Explain why $f(g)=g^{m_{f}}$ for some $m_{f}\in\mathbb{Z}_{n}$.

3. Show that the $m_{f}$ obtained in (ii) is unique to $f$.

4. Suppose the functions $f_{1}:G\to G$ and $f_{2}:G\to G$ are homomorphisms. Prove that

   $$m_{f_{1}\circ f_{2}}=m_{f_{1}}\otimes_{n}m_{f_{2}}$$

   where $\circ$ denotes function composition and $\otimes_{n}$ denotes multiplication modulo $n$.

5. Let the function $f:G\to G$ be a homomorphism. Prove that $f$ is an automorphism if and only if $m_{f}$ has a multiplicative inverse modulo $n$. That is, there exists $k\in\mathcal{U}_{n}$ such that $m_{f}k\equiv 1\pmod{n}$ if and only if $f$ is an automorphism.(Hint: an exercise given elsewhere in the book states that $ab\equiv 1\pmod{m}$ if and only if $\gcd(a,m)=1$.)

6. Hence, by considering the map $\phi:\mathrm{Aut}(G)\to\mathcal{U}_{n}$ where $\phi(f)=m_{f}$, prove that

   $$\mathrm{Aut}(G)\cong\mathcal{U}_{n}.$$