Is it true that for every finite cyclic group <G,*> with generator a each of the unique elements of the group may be presented as $\displaystyle a^n$ where n is non-negative integer.

My proof: If G is finite, there is some positive integer m for which $\displaystyle a^m=e$. Then $\displaystyle a^{m+1}=a^1$, $\displaystyle a^{m+2}=a^2$ etc...

Moreover $\displaystyle a^m=a^{-m}=e$. Then $\displaystyle a^m*a^1=a^{-m}*a^1$. Therefore $\displaystyle a^{m+1}=a^{-m+1}=a^1$.

If $\displaystyle a^{m+n}=a^{-m+n}$ for some n<m then $\displaystyle a^{m+n+1}=a^{-m+n+1}$ too.

This must cover all inverses of the elements from $\displaystyle a^1$ to $\displaystyle a^m$.