Proving The Fundamental Homomorphism Theorem | Abstract Algebra

We introduce and prove the fundamental homomorphism theorem (also called the first isomorphism theorem), which states that every homomorphic image of a group G is isomorphic to the quotient group of G by the kernel of f, given that f is a homomorphism from G onto H. To prove this we will use some prior results. We previously proved that a quotient group of G is a homomorphic image of G, so we now have that a group is a quotient group of G if and only if it is a homomorphic image of G (or isomorphic to one). So homomorphic images and quotient groups are somewhat interchangeable. #abstractalgebra #grouptheory

Proof f(a)=f(b) iff Ka=Kb:    • A Kernel Theorem: f(a)=f(b) iff Ka=Kb...  
Normal Subgroups:    • Definition of Normal Subgroups | Abst...  
Coset Multiplication on Normal Subgroups:    • Coset Multiplication on Normal Subgro...  
Kernels of Homomorphisms:    • Kernels of Homomorphisms | Abstract A...  
Quotient Groups and Homomorphic Images:    • Quotient Groups and Homomorphic Image...  

Abstract Algebra Course:    • Abstract Algebra  
Abstract Algebra Exercises:    • Abstract Algebra Exercises  

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