Consider $G$, a finite abelian group where $\forall a\in G, a\neq e$, we have $a^2 \neq e$. Given that the list $a_1,...,a_n$ lists out all elements of $G$ without repetition, evaluate $a_1...a_n$ (their product).
To me this seems like a bit of an oddly phrased question, unless my intuition is correct.
My intuition: because $G$ is abelian, rearrange the elements in the list so that each element is adjacent to its own inverse. Then, it is clear that $a_1...a_n=e$.
Could it really be this simple? Is there a more rigorous way to phrase my argument?