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:

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.