proving the Klein 4 group is abelian 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?

You don't need to prove it for this particular group, because ALL 4-element groups are abelian, you can prove that like this:
Suppose that for all $a\in G$, we have that $a^2=1$, or equivalently $a=a^{-1}$, then the group is abelian:
$$(ab)=(ab)^{-1}=b^{-1}a^{-1}=ba$$
If our supposition is not true, then there exists an element $a$ such that $a^2\neq 1$ different from $1$, and the group generated by this element, by Lagrange's theorem, must have order $4$, so $\langle a\rangle=G$, so the group is cyclic and abelian.
