How do we determine if a subgroup of $S_n$ is normal?

Question

Ok, so I have $H = \{ e,(12),(34),(12)(34) \}$ being a subgroup of $S_4$. How do I prove that it is also normal (or isn't)? Because I don't really want to check every case, and there must be a more elegant way.

Thanks in advance.

Answer 1

Elements in $S_n$ are conjugate if and only if they have the same cycle structure. In particular, $(12)$ and $(13)$ are conjugate, so since $(13) \not \in H$ then $H$ is not a normal subgroup of $S_4$.

$S_4$ does have a normal subgroup isomorphic to $H$, but it is generated by the permutations having cycle structure as follows: $(\cdot \cdot) (\cdot \cdot)$.

Answer 2

Any two elements with the same cycle structure are conjugate. Thus, your subgroup $H$ is not inner automorphism invariant, and so not normal. For instance, $(12)(34)$ and $(13)(24)$ are conjugate.