Show that $G/H\cong S_3$

Solution 1:

Here is a way to construct an homomorphism from $S_4$ to $S_3$, which has kernel $H$. Consider the three possible partitions of the set $\{1,2,3,4\}$ into two sets:

$$A=\{1,2\}\cup\{3,4\}; \quad B=\{1,3\}\cup \{2,4\}; \quad C=\{1,4\} \cup \{2,3\}.$$

Now you can see what does an element $p\in S_4$ to those partitions. For instance the element $p := (1234)$ does the following: \begin{align*} \{1,2\} &\overset{p}{\leadsto} \{2,3\}; &\{1,3\}\overset{p}{\leadsto}\{2,4\};\\\{1,4\}&\overset{p}{\leadsto} \{2,1\}; &\{2,3\}\overset{p}{\leadsto}\{3,4\}; \\ \{2,4\} &\overset{p}{\leadsto}\{3,1\}; &\{3,4\}\overset{p}{\leadsto}\{4,1\}, \end{align*}

so $$A \overset{p}{\leadsto}\{2,3\}\cup \{4,1\}; \quad B \overset{p}{\leadsto} \{2,4\}\cup \{1,3\}; \quad C \overset{p}{\leadsto} \{2,1\}\cup \{3,4\}.$$ This corresponds to the transposition $[ABC]\overset{p}{\leadsto}[CBA]$. Which is like the transposition $(13)$ in $S_3$. Now define the map $$\varphi:S_4\to S_3,$$

which sends an element of $S_4$ to the action on the partitions $A,B,C$. If $q,r \in S_4$ then the product $qr$ corresponds to the composition of the action of $q$ and of $r$, so the action of $qr$ is the result of the two actions, that is $\varphi(qr)=\varphi(q)\varphi(r)$ and so you have an homomorphism. Moreover you can see that the only elements which sends $[ABC]$ to $[ABC]$ (written in one line notation) are exactly the elements of $H$, so $H$ is the kernel of $\varphi$ and therefore $S_4/H\approx S_3$