# The Internal Direct Product of Two Groups

Recall from The External Direct Product of Two Groups page that if $(G, *)$ and $(H, +)$ are two groups then the external direct product of these groups is the new group $(G \times H, \cdot)$ where $\cdot$ is the binary operation defined for all $(g_1, h_1), (g_2, h_2) \in G \times H$ by:

(1)We will now define another type of "direct product" known as the internal direct product of two groups.

Definition: Let $(G, *)$ be a group and let $(H, *)$ and $(K, *)$ be two subgroups of $(G, *)$. Then $(G, *)$ is said to be the Internal Direct Product of $(H, *)$ and $(K, *)$ if:1) $G = \{ h * k : h \in H, k \in K \}$.2) $H \cap K = \{ e \}$ where $e$ is the identity element in $G$.3) $h * k = k * h$ for all $h \in H$ and for all $k \in K$. |

For example, consider the group $(\mathbb{Z}_6, +)$ and the following subgroups:

(2)Note that $\{ h * k : h \in H, k \in K \} = \{ 0, 1, 2, 3, 4, 5 \} = G$, so the first condition is met. Also the identity for $\mathbb{Z}_6$ is $e = 0$ and $H \cap K = \{ 0 \}$ so the second condition is met. Lastly $\mathbb{Z}_6$ is an abelian group so the third condition is met.