Internal Direct Product Theorem/Examples/Multiplicative Monoid of Integers Modulo 6

Example of Use of Internal Direct Product Theorem
Consider the multiplicative monoid of integers modulo $6$ $\struct {\Z_6, +_6}$, illustrated by Cayley Table:

Let $H := \set {0, 2, 4}$.

Let $K := \set {0, 3}$.

We have that:
 * $H \times_6 K = \set 0$

so $\struct {\Z_6, \times_6}$ is not the internal group direct product of $H$ and $K$.