Symmetric Group on 3 Letters/Centralizers

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

The centralizers of each element of $S_3$ are given by:

Proof
By definition, the centralizer of $a$ (in $S_3$) is defined as:


 * $\map {C_{S_3} } a = \set {x \in S_3: x \circ a = a \circ x}$

The centralizer are then apparent by inspection of the Cayley table.