second isomorphism theorem

Let $(G,*)$ be a group. Let $H$ be a subgroup of $G$ and let $K$ be a normal subgroup of $G$ . Then

•
$HK:=\\{h*k\\mid h\\in H,\\ k\\in K\\}$ is a subgroup of $G$ ,
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$K$ is a normal subgroup of $H K$ ,
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$H\\cap K$ is a normal subgroup of $H$ ,
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There is a natural group isomorphism $H/(H\\cap K)=HK/K$ .

The same statement also holds in the category of modules over a fixed ring (where normality is neither needed nor relevant), and indeed can be formulated so as to hold in any abelian category.

单词	SecondIsomorphismTheorem
释义	second isomorphism theorem Let $(G,)$ be a group. Let $H$ be a subgroup of $G$ and let $K$ be a normal subgroup of $G$ . Then • $HK:=\\{hk\\mid h\\in H,\\ k\\in K\\}$ is a subgroup of $G$ , • $K$ is a normal subgroup of $H K$ , • $H\\cap K$ is a normal subgroup of $H$ , • There is a natural group isomorphism $H/(H\\cap K)=HK/K$ . The same statement also holds in the category of modules over a fixed ring (where normality is neither needed nor relevant), and indeed can be formulated so as to hold in any abelian category.
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