Check whether subgroup is normal in $ S_4$

Question

Let H={e,(12)(34)} and K ={e, (12)(34), (13)(24), (14)(23)} be subgroups of $ S_4$ where e is identity element. Then which of following is true

1. H and K are normal subgroups of $S_4$

2. H is normal in K and K is normal in$ A_4$

3. H is normal in $A_4$, but not normal in $ S_4$

4. K is normal in $S_4$ but H is not

How should this question be approached, checking left and right cosets of these subgroup is very time consuming. Checking gh$ g^{-1}$ $\in $ H for every g in $ S_4$ is also time consuming.

Answer 1

Two elements in $S_n$ are conjugated if and only if they have the same factoriziation to disjoint cycels.

Hence $K$ is normal in $S_4$ and therefore in $A_4$.

From the same reason $H$ is not normal in $S_4$.

$H$ is normal in $K$ since it is a subgroup of index $2$.

I left for you to cheack if $H$ is normal in $A_4$.