# What does it mean to evaluate the product of elements of a group?

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?

• You're almost there. But why is there no element which is it's own inverse? In that case your reasoning doesn't work. Jan 23, 2017 at 22:15
• math.stackexchange.com/questions/53185/… Jan 23, 2017 at 22:21

This argument is correct. The assumption $a^2\ne e$ for all $a\ne e$ tells us that the inverse of every $a\ne e$ differs from $a$, so the construction of your list is possible.
Obviously this assumption is essential. Indeed, in $\Bbb Z_4=\{0,1,2,3\}$ (with addition) we have $0+1+2+3=2$. In this group we have $2+2=0$.