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
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $5$: Subgroups: Exercise $16$