Symmetric Group on 3 Letters/Normalizers

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

The normalizers of each subgroup of $S_3$ are given by:

Proof
The subgroups of $S_3$ are as follows:

By Normalizer of Subgroup is Largest Subgroup containing that Subgroup as Normal Subgroup:


 * $\map {N_{S_3} } {\set e}$ is the largest subgroup $N$ of $S_3$ in which $H$ is normal in $N$.

From Trivial Subgroup is Normal:
 * $\map {N_{S_3} } {\set e} = S_3$

From Group is Normal in Itself:
 * $\map {N_{S_3} } {S_3} = S_3$

The index of $\set {e, \tuple {123}, \tuple {132} }$ is $2$.

Hence from Subgroup of Index 2 is Normal, $\set {e, \tuple {123}, \tuple {132} }$ is normal in $S_3$.

It follows that:
 * $\map {N_{S_3} } {\set {e, \tuple {123}, \tuple {132} } } = S_3$

From Normal Subgroups of Symmetric Group on 3 Letters, none of $\set {e, \tuple {12} }$, $\set {e, \tuple {13} }$ and $\set {e, \tuple {23} }$ is normal in $S_3$.

There are no larger subgroup of $S_3$ containing any of them, so they are their own normalizers.