Symmetric Group on 3 Letters/Subgroups/Examples/Non-Subgroup

Example of Subset of Symmetric Group on 3 Letters which is not a Subgroup
Let $S_3$ denote the Symmetric Group on $3$ Letters, whose Cayley table is given as:

Consider the subset $H$ of $S_3$:


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

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

Proof
We have:
 * $\tuple {12} \circ \tuple {13} = \tuple {123}$

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

Thus $H$ is not closed under $\circ$.

Hence the result by definition of subgroup.