[Math] Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.

Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.

This question is from group theory in Abstract Algebra and no matter how many times my lecturer teaches it for some reason I can't seem to crack it.

(Please note that $e$ in the question is the group's identity.)

Here's my attempt though…

First I understand Abelian means that if $g_1$ and $g_2$ are elements of a group $G$ then they are Abelian if $g_1g_2=g_2g_1$…

So, I begin by trying to play around with the elements of the group based on their definition…

$$(g_2g_1)^r=e$$
$$(g_2g_1g_2g_2^{-1})^r=e$$
$$(g_2g_1g_2g_2^{-1}g_2g_1g_2g_2^{-1}…g_2g_1g_2g_2^{-1})=e$$

I assume that the $g_2^{-1}$'s and the $g_2$'s cancel out so that we end up with something like,

$$g_2(g_1g_2)^rg_2^{-1}=e$$
$$g_2^{-1}g_2(g_1g_2)^r=g_2^{-1}g_2$$

Then ultimately…

$$g_1g_2=e$$

I figure this is the answer. But I'm not totally sure. I always feel like I do too much in the pursuit of an answer when there's a simpler way.

Reference: Fraleigh p. 49 Question 4.38 in A First Course in Abstract Algebra.