Symmetric Group on 3 Letters/Normal Subgroups

Normal Subgroups of the Symmetric Group on 3 Letters
Let $S_3$ denote the Symmetric Group on 3 Letters, whose Cayley table is given as:

Consider the subgroups of $S_3$:

Of those, the normal subgroups in $S_3$ are:
 * $S_3, \set e, \set {e, \tuple {123}, \tuple {132} }$

Proof
$S_3$ itself is normal in $S_3$ by Group is Normal in Itself.

$\set e$ is normal in $S_3$ by Trivial Subgroup is Normal.

$\set {e, \tuple {12} }$:

Hence $\set {e, \tuple {12} }$ is not normal in $S_3$.

$\set {e, \tuple {23} }$:

Hence $\set {e, \tuple {23} }$ is not normal in $S_3$.

$\set {e, \tuple {13} }$:

Hence $\set {e, \tuple {13} }$ is not normal in $S_3$.

$\set {e, \tuple {123}, \tuple {132} }$:

We have that $\set {e, \tuple {123}, \tuple {132} }$ is the set of even permutations of $S_3$.

Any permutation of the form $\alpha \pi \alpha^{-1}$, for $\pi$ even, is also even.

Thus:
 * $\forall \alpha \in S_3: \alpha \pi \alpha^{-1} \in \set {e, \tuple {123}, \tuple {132} }$

Hence $\set {e, \tuple {123}, \tuple {132} }$ is normal in $S_3$.