Symmetric Group on 3 Letters/Order of Elements

Orders of Elements 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 orders of the various elements of $S_3$ are:

\(\ds e: \, \)

\(\ds \)

\(\)

\(\ds \)

Order $1$

\(\ds \tuple {123}: \, \)

\(\ds \tuple {123}^2\)

\(=\)

\(\ds \tuple {132}\)

\(\ds \tuple {123} \tuple {132}\)

\(=\)

\(\ds e\)

hence Order $3$

\(\ds \tuple {132}: \, \)

\(\ds \tuple {132}^2\)

\(=\)

\(\ds \tuple {123}\)

\(\ds \tuple {132} \tuple {123}\)

\(=\)

\(\ds e\)

hence Order $3$

\(\ds \tuple {12}: \, \)

\(\ds \tuple {12}^2\)

\(=\)

\(\ds e\)

hence Order $2$

\(\ds \tuple {13}: \, \)

\(\ds \tuple {13}^2\)

\(=\)

\(\ds e\)

hence Order $2$

\(\ds \tuple {23}: \, \)

\(\ds \tuple {23}^2\)

\(=\)

\(\ds e\)

hence Order $2$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Exercise $4$