group homomorphism of permutation groups

group homomorphism of permutation groups

Suppose I let $g_{1} = y_{1}y_{2} + y_{3}y_{4}$, $g_{2} = y_{1}y_{3} +
y_{2}y_{4}$, and $g_{3} = y_{1}y_{4} + y_{2}y_{3}$. The group $S_{4}$
(permutations of the set $\{1,2,3,4\}$), which acts on the set
$\{y_{1},y_{2},y_{3},y_{4}\}$. This yields a permutation of the set
$\{g_{1},g_{2},g_{3}\}$. This is a group homomorphism (call it $f$) from
$S_{4}$ to $S_{3}$.
There are two things I am curious about. What are all the elements in the
kernel of $f$? Also, after doing physical examples with this, I am pretty
sure the ker $f$ is an Abelian group, although I am unsure how to prove
it.