Symmetric Group on 3 Letters/Subgroups

Subgroups 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 subsets of $S_3$ which form subgroups of $S_3$ are:

\(\ds \)

\(\)

\(\ds S_3\)

\(\ds \)

\(\)

\(\ds \set e\)

\(\ds \)

\(\)

\(\ds \set {e, \tuple {123}, \tuple {132} }\)

\(\ds \)

\(\)

\(\ds \set {e, \tuple {12} }\)

\(\ds \)

\(\)

\(\ds \set {e, \tuple {13} }\)

\(\ds \)

\(\)

\(\ds \set {e, \tuple {23} }\)

Examples

Non-Subgroup

Consider the subset $H$ of $S_3$:

$H = \set {e, \tuple {12}, \tuple {13}, \tuple {23} }$

Then $H$ is not a subgroup of $S_3$.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.2$. Subgroups: Example $93$
• 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): $1$: Subgroups: Problem $1.1$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Exercise $2$