Union of Subgroups/Examples/Subgroups of S3

Examples of Union of Subgroups

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

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

Consider the subgroups $H, K \le G$:

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

$K = \set {e, \tuple {13} }$

We have that:

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

and:

$\tuple {12} \circ \tuple {13} = \tuple {123}$

But $\tuple {123} \notin H \cup K$.

Hence $H \cup K$ is not closed and so is not a group.

The result follows by definition of subgroup.

$\blacksquare$

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{J}$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Exercise $2$