Isomorphism and Correspondence Theorems
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Abstract AlgebraHomomorphisms and Isomorphisms of GroupsIsomorphism and Correspondence Theorems
Homomorphic Images and Normal Subgroups
Images Controlled by Normal Subgroups
The first isomorphism theorem shows that every epimorphism from produces a quotient of . The focus keyword homomorphic images and normal subgroups captures this exact connection: homomorphic images of a group are the same, up to isomorphism, as quotient groups by normal subgroups. If is onto, then , and the kernel is normal in . Conversely, every normal subgroup gives a natural homomorphism onto a quotient group. This lesson turns the first isomorphism theorem into a classification principle for possible epimorphic images.
DEFINITION : Homomorphic Image
Let and be groups. The group is called a \textbf{homomorphic image} of if there exists an epimorphism .
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Author
Dr. Bivash Majumder
Assistant Professor in Mathematics
Prabhat Kumar College, Contai