Normal Klein four-subgroup of symmetric group S4

Question

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one of them is the Normal Klein four-subgroup.

I don't understand this. According this article, $H = \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$. Is $H$ a subgroup at all? $(12)$ sends $1234$ to $2134$, $(23)$ sends $2134$ to $2314$, but no element of $H$ sends $1234$ to $2314$. Do I misunderstand something?

Answer 1

You've misread the notation. (12) is not an element of H - (12)(34) is. It sends 1234 to 2143. Then (14)(23) sends that to 3412, which is also (13)(24).