Understanding a Cyclic Group Proof

Question

For one of my questions, the following answer was given by Suzet:

The thing is: I do not see why... Is it because of inverses or because having a negative could make $i-j$ positive? I asked this in the comment but no one replied yet. Does anyone know? :) I feel like I am missing something very trivial.

Answer 1

A variant: $\enspace$if $a^{-n}=e\;(n\in\mathbf N)$, then $\; e=e^{-1}=(a^{-n})^{-1}=a^{(-n)(-1)}=a^n.$

Answer 2

Suppose that for a positive integer $n$, we have $a^{-n} = e$. Then multiplying by $a$ on each side, we get $a = a^{-n+1}$. Then we simply have: $$ a^n = (a^{-n+1})^n = a^{-n^2+n} = a^{-n(n-1)} = (a^{-n})^{n-1} = e^{n-1} = e $$