Symmetric Group on 3 Letters/Centralizers

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Centralizers of the Symmetric Group on 3 Letters

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

$\begin{array}{c|cccccc} \circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$


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

\(\ds \map {C_{S_3} } e\) \(=\) \(\ds S_3\)
\(\ds \map {C_{S_3} } {123}\) \(=\) \(\ds \set {e, \tuple {123}, \tuple {132} }\)
\(\ds \map {C_{S_3} } {132}\) \(=\) \(\ds \set {e, \tuple {123}, \tuple {132} }\)
\(\ds \map {C_{S_3} } {12}\) \(=\) \(\ds \set {e, \tuple {12} }\)
\(\ds \map {C_{S_3} } {23}\) \(=\) \(\ds \set {e, \tuple {23} }\)
\(\ds \map {C_{S_3} } {13}\) \(=\) \(\ds \set {e, \tuple {13} }\)


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.

$\blacksquare$


Sources