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

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$HK:=\{h*k\mid h\in H,\ k\in K\}$ 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.

Title	second isomorphism theorem
Canonical name	SecondIsomorphismTheorem
Date of creation	2013-03-22 12:08:46
Last modified on	2013-03-22 12:08:46
Owner	djao (24)
Last modified by	djao (24)
Numerical id	9
Author	djao (24)
Entry type	Theorem
Classification	msc 13C99
Classification	msc 20A05