Identity element in a finite cyclic group $G$.
Solution 1:
The clue is in the name: it is cyclic. There is no order to the elements. But if you look at all powers of $a$, you will get $a^{i+kn}=a^i$ for all $k\in\Bbb Z$, $i\in\{0,\dots,n-1\}$, meaning that the powers are in a cycle. Note that $a^0=e$ by definition, but $0$ is not positive.