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 $H K$ ,
•
$H\\cap K$ 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.

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.
随便看	WellorderingPrincipleImpliesAxiomOfChoice well-ordering principle implies axiom of choice WellQuasiOrdering well quasi ordering WeylAlgebra Weyl algebra WeylChamber Weyl chamber WeylGroup Weyl group WeylMinkowskiTheorem Weyl-Minkowski theorem WeylsCriterion WeylsInequality WeylsTheorem Weyl’s criterion Weyl’s inequality Weyl’s theorem WheelFactorization wheel factorization WheelGraph wheel graph WhenAllSingularitiesArePoles when all singularities are poles WhenAreBallsSeparated