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Homesecond isomorphism theorem
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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 $G$,

$K$ is a normal subgroup of $HK$,

$H\cap K$ is a normal subgroup of $H$,

There is a natural group isomorphism $H/(H\cap K)=HK/K$.
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Theorem
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