Identify the following elements in $S_4$ modulo $H = \{e, (12)(34), (13)(24), (14)(23)\}$

Question

Identify the following elements in $S_4$ modulo $H =\{e, (12)(34), (13)(24), (14)(23)\}$

$(1234)$

$(12)(1324)$

$(14)(1234)$

I see that $H$ is a normal subgroup. But I don't clearly understand what the problem is asking about.

The options I can choose from are $(123), (12), e$

I think $S_4/H$ is isomorphic to $S_3$ and the problem wants me to find what each element above matches to in $S_3$?

Answer 1

Note that\begin{align}(1\ \ 2\ \ 3)H&=\{(1\ \ 2\ \ 3),(1\ \ 2\ \ 3)(1\ \ 2)(3\ \ 4),(1\ \ 2\ \ 3)(1\ \ 3)(2\ \ 4),(1\ \ 2\ \ 3)(1\ \ 4)(2\ \ 3)\}\\&=\{(1\ \ 2\ \ 3),(1\ \ 3\ \ 4),(2\ \ 4\ \ 3),(1\ \ 4\ \ 2)\}.\end{align}On the other hand$$(1\ \ 2)(1\ \ 3\ \ 2\ \ 4)=(1\ \ 3)(2\ \ 4)\quad\text{and}\quad(1\ \ 4)(1\ \ 2\ \ 3\ \ 4)=(1\ \ 2\ \ 3).$$So, $(1\ \ 2)(1\ \ 3\ \ 2\ \ 4)\notin(1\ \ 2\ \ 3)H$, whereas $(1\ \ 4)(1\ \ 2\ \ 3\ \ 4)\in(1\ \ 2\ \ 3)H$. In particular, $(1\ \ 4)(1\ \ 2\ \ 3\ \ 4)$ and $(1\ \ 2\ \ 3)$ are the same element modulo $H$.