I've seen similar proofs which rely on Lagrange's theorem which I'm not allowed to use. Other proofs just showed that there exists some exponent that satisfies the equation.
In my case I want to show (1) that for all $a \in G:a^n=e$, where $n=|G|$ and $e$ is the neutral element of the finite abelian group $G$ and (2) there exists a natural number $k$ so that $a^k=e$ (this k should be the order of the element $a$ noted as $k=ord_G(a)$.
Unfortunately I also count't find any online material. Any help would be appreciated!