---- > [!definition] Definition. ([[internal direct product of subgroups]]) > Let $H$ and $K$ be [[subgroup]]s of a [[group]] $G$. The following are equivalent: > 1. $H$ and $K$ are [[normal subgroup|normal]] in $G$ and every element in $G$ can be written *uniquely* as a product $hk$, with $h \in H$ and $k \in K$; > 2. $H$ and $K$ are [[normal subgroup|normal]] in $G$, $G=HK$, and $H \cap K = (e).$; > 3. The natural map $H \times K \to G$, $(h,k) \mapsto hk$ is a [[group isomorphism]]. > \ > In this case we say that $G$ is the **internal direct product** of the ([[normal subgroup|normal]]) [[subgroup]]s $H$ and $K$. ^767c9c > [!justification] > We will show $(1) \implies (2) \implies (3) \implies 1$. ># $(1) \implies (2)$ Suppose $H,K \trianglelefteq G$ and every $g \in G$ can be written uniquely as a product $hk$, with $h \in H$ and $k \in K$. Then by assumption $G$ and $H$ are [[normal subgroup|normal]] with $G=HK$. That $H \cap K= (e)$ is clear too: if not we could represent $a \in H \cap K$ as both $a = \overbrace{a}^{\in H} \overbrace{e}^{\in K} \text{ and } a = \overbrace{e}^{\in H} \overbrace{a}^{\in K},$ and $a$ would not factor uniquely. ># $(2) \implies (3)$ This is shown in [[external-internal group products are isomorphic]]. ># $(3) \implies (1)$ Since the [[group isomorphism]] $\phi:H \times K \to G$ mapping $(h,k) \mapsto hk$ is a [[surjection]], it is clear that every element in $G$ can be written as a product $hk$. Since it is an [[injection]], it is clear that said factorization is unique. > [!note] Remark. > Above we show that the existence of [[internal direct product of subgroups|internal direct product]] implies existence of [[direct product of groups|external direct product]] . It is also true that the existence of external $\implies$ existence of internal direct products. > \ > Ultimately, the relation between [[internal direct product of subgroups|internal]] and [[direct product of groups|external]] direct products of [[subgroup]]s is that [[external-internal group products are isomorphic]]. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```