I am doing some exercises in Algebra: Chapter 0. In the second chapter, we are asked to prove the following:
$G$ is a finite group with a unique element $f$ of order $2$. Then $\operatorname{\Pi_{g\in G}}g=f$.
This result is highly plausible. If we multiply the elements in the order of \begin{equation}e\cdot f\cdot \text{elements of order 3}\cdot\text{elements of order 4}\cdots,\end{equation} and pair elements with their inverses, then we get $f$, since it is the only element that does not have a couple.
However this is only one possible order of multiplication, and we know that in general different order give different results.
So I wonder how we can do the general case. Thanks!