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 $HK$,

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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  20130322 12:08:46 
Last modified on  20130322 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 