Prove that a group where $a^2=e$ for all $a$ is commutative Defining a group $(G,*)$ where $a^2=e$ with $e$ denoting the identity class....
I am to prove that this group is commutative. To begin doing that, I want to understand what exactly the power of 2 means in this context. Is the function in the group a power or something?

If $a^2 =e$ for all $a\in G$, then $a^{-1}(a^2) = a^{-1}e = a^{-1}$ for all $a\in G$, but since $a^{-1}(a^2) = (a^{-1}a)a =ea = a$, we have $a^{-1} = a$ for all $a \in G$.  If we now take any $a, b \in G$, we see that $(ab)^{-1} = ab$.  But $(ab)^{-1} = b^{-1}a^{-1} = ba$; thus $ab = ba$ for all $a,b \in G$ and so $G$ is commutative. QED.
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
