There have been many answers to this question but I wanted to find a new approach to the problem without using the isomomorphism theorems.
By some explicit calculations I have shown that $V$ can be mapped to $e$ and the corresponding cosets can be mapped to all the elements of order $2$,and the elements order $3$ and so on. Now what I wanted to prove that such a map is an homomorphism. Is there a way I can use matrices to prove the homorphism.
To sumup:
I have shown that ${(12)(34),(14)(23),(13)(24),1}$ can be mapped to $e$ and the elements of the form $(12)V,(23)V,(13)V,(123)V,(132)V$ can be mapped to $s,sr,sr^2,r,r^2$.Next what I want to prove is that such a map is a homorphism. How do I show that( using matrices preferably)or any other method if there exist except the isomorphism theorems?
