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?

There are only $4$ elements, so just check by hand that $ab = ba$ for any $a, b$.  It's true automatically if either $a$ or $b$ is $1$ so actually there are only three pairs to check.
