Internal Direct Product Theorem/Examples/Additive Group of Integers Modulo 6

Example of Use of Internal Direct Product Theorem
Consider the additive group of integers modulo $6$ $\struct {\Z_6, \times_6}$, illustrated by Cayley Table:

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

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

We have that:
 * $H +_6 K = \struct {\Z_6, +_6}$

and:
 * $H \cap K = \set 0$

Hence $H$ and $K$ are subgroups of $\struct {\Z_6, +_6}$ which fulfil the conditions of the Internal Direct Product Theorem.

Thus $\struct {\Z_6, +_6}$ is the internal group direct product of $H$ and $K$.

Because:
 * $H$ is isomorphic to $\struct {\Z_3, +_3}$
 * $K$ is isomorphic to $\struct {\Z_2, +_2}$

it follows by Isomorphism of External Direct Products that:
 * $\struct {\Z_6, +_6}$ is isomorphic to $\struct {\Z_3, +_3} \times \struct {\Z_2, +_2}$.