Encyclopedia > Fundamental theorem on homomorphisms
In abstract algebra, for a number of algebraic structures, the fundamental theorem on homomorphisms relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
Let G and H be groups; let f : G->H be a group homomorphism; let K be the kernel of f; let φ be the natural surjective homomorphism G->G/K. Then there exists a unique homomorphism h:G/K->H such that f = h φ. Moreover, h is injective and provides an isomorphism between G/K and the image of f.
In abstract algebra, the fundamentaltheorem on homomorphisms, also known as the fundamentalhomomorphismtheorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.
The homomorphismtheorem is used to prove the isomorphism theorems.
Given two groups G and H and a grouphomomorphism f : G→H, let K be a normal subgroup in G and φ the natural surjectivehomomorphism G→G/K.
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