Internal direct product

Definition

Let $G$ be a group and let $H$ and $K$ be normal subgroups of $G$. The group $G$ is the internal direct product of $H$ and $K$ when $H \cap K = \{e\}$ and $HK = G$.

Remarks

In this case, $hk = kh$ for all $h \in H$ and $k \in K$, because $hkh^{-1}k^{-1}$ lies in $H \cap K$. Then the map $(h, k) \mapsto hk$ is an isomorphism from the direct product $H \times K$ to $G$. Also, each element of $G$ is a product $hk$ with $h \in H$ and $k \in K$ in exactly one way. For example, $\mathbb{Z}/6\mathbb{Z}$ is the internal direct product of $\{0, 3\}$ and $\{0, 2, 4\}$.

Explanations

L1 human

Sometimes a group already contains two subgroups that behave like the two factors of a direct product. They meet only in the identity, they fill the group together, and they are both normal. Then the group is the direct product of these subgroups, up to isomorphism.