We are asked to find smallest normal subgroup of $S_4$ which contains $\langle(1,3,2,4)\rangle = H$.
I know that a subgroup $G$ is normal if:
$$\forall x \in S_4, xH = Hx$$
I know that $H$ contains at least $4$ elements generated by $\langle(1,3,2,4)\rangle$. I don't know, however, how should I know which elements should be added from $S_4$.
Best Answer
We already know the nontrivial proper normal subgroups of $S_4$, namely $A_4$ and $V_4$. Since $V_4$ has only elements of order $2$ and $1$, and $(1324)$ is not in $A_4$, the smallest normal subgroup containing $H$ must be $S_4$.
Reference:
A question on identifying normal subgroups of $S_4$
how do I prove that $S_4$ has no normal subgroup of order 6