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**Danneedshelp** Theorem: for any positive integer n, $\displaystyle Aut(Z_{n})$ is isomorphic to $\displaystyle U(n)$.

Q: Prove that every element $\displaystyle g\in\\Aut(Z_{n})$ is determined by $\displaystyle g(1)$; meaning, $\displaystyle g(1)$ results in a well-defined automorphism of $\displaystyle Z_{n}$.

A: let $\displaystyle g\in\\Aut(Z_{n})$ be arbitrary. Then $\displaystyle f:Z_{n}\rightarrow\Z_{n}$ is an isomorphism. So, to show if something is well-defined I need to show if a=c, b=d, then a*b=c*d. I am not sure what to my a,b, and c's are. I know $\displaystyle f(k)=kf(1)$. so, do I let $\displaystyle f(m)=mf(1)$ and $\displaystyle f(p)=pf(1)$. Now, I want to show $\displaystyle f(p)*f(q)=mf(1)*pf(1)$?

Thanks