TITLE: proving the Klein 4 group is abelian
QUESTION [6 upvotes]: If ker $f = \{ (1), (12)(34), (14)(23), (13)(24) \}$, which is the Klein 4 group, how do I prove that it is an Abelian group? Do I just show that each element in the ker $f$ is commutative to prove this?

REPLY [3 votes]: This is a slightly less direct approach--one you probably would not think of if you were attempting a problem like this for the first time)--but hopefully you (or others) will find this alternative solution to be instructive.
Theorem: Let $G$ be a group such that every element of $G$ other than the identity has order $2$. Then $G$ is abelian.
Proof: Let $x$ and $y$ be any two elements of $G$. Then we must show that $xy=yx$. I will write $e$ to indicate the identity element.
$$e=(xy)^2=xyxy$$
$$x^{-1}=x^{-1}e=x^{-1}xyxy=yxy$$
$$y^{-1}x^{-1}=y^{-1}yxy=xy$$
Now note that since $x$ and $y$ each have order $2$, $x=x^{-1}$ and $y=y^{-1}$, so $yx=xy$.
I leave it up to you to verify that the non-identity elements of the group in question have order $2$.