Chapter 3.3, Problem 40E

Find subgroups $H$ and $K$ of the group $S(A)$ in example $3$ of the section $3.1$ such that the set $HK$ defined in Exercise $39$ is not a subgroup of

$S(A)$.

From Example $3$ of section $3.1$: $A=\{1,2,3\}$ and $S(A)$ is a set of all permutations defined on $A$.

$HK$ defined in Exercise $39$: $HK=\left\{g\in G|g=hkforh\in Handk\in K\right\}$