Symmetric Group on 3 Letters/Generators

From ProofWiki
Jump to navigation Jump to search

Generators of the Symmetric Group on 3 Letters

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

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


Let:

\(\ds G_1\) \(=\) \(\ds \set {\tuple {123}, \tuple {12} }\)
\(\ds G_2\) \(=\) \(\ds \set {\tuple {13}, \tuple {23} }\)


Then:

\(\ds S_3\) \(=\) \(\ds \gen {G_1}\)
\(\ds \) \(=\) \(\ds \gen {G_2}\)

where $\gen G$ denotes the group generated by a subset $G$ of $S_3$.


Proof

For $G_1$:

\(\ds e\) \(=\) \(\ds \tuple {12} \tuple {12}\)
\(\ds \tuple {123}\) \(=\) \(\ds \tuple {123}\)
\(\ds \tuple {132}\) \(=\) \(\ds \tuple {123} \tuple {123}\)
\(\ds \tuple {12}\) \(=\) \(\ds \tuple {12}\)
\(\ds \tuple {23}\) \(=\) \(\ds \tuple {123} \tuple {12}\)
\(\ds \tuple {13}\) \(=\) \(\ds \tuple {12} \tuple {123}\)


For $G_2$:

\(\ds e\) \(=\) \(\ds \tuple {23} \tuple {23}\)
\(\ds \tuple {123}\) \(=\) \(\ds \tuple {13} \tuple {23}\)
\(\ds \tuple {132}\) \(=\) \(\ds \tuple {23} \tuple {13}\)
\(\ds \tuple {12}\) \(=\) \(\ds \tuple {13} \tuple {23} \tuple {13}\)
\(\ds \tuple {23}\) \(=\) \(\ds \tuple {23}\)
\(\ds \tuple {13}\) \(=\) \(\ds \tuple {13}\)

$\blacksquare$


Sources