Symmetric Group on 3 Letters/Generators

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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

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.3$. Subgroup generated by a subset: Example $97$
1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $5$: Subgroups: Exercise $8$
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Example $4.9$