Math 627 A : Modern Algebra I Notes on the Isomorphism Theorems for Groups 1 Fundamental Results : Up to the First Isomorphism Theorem
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چکیده
A brief comment about items (3) and (4). If H is a subgroup of G the inclusion map is an injective homomorphism from H to G . On the other hand, the image of an injective homomorphism α : H −→ G is a subgroup that is isomorphic to H . So the study of injective homomorphisms is (up to isomorphism) the study of subgroups. After studying subgroups, we defined normal subgroup and showed several equivalent properties determining that a subgroup is normal. If N is normal in G , we can define a quotient group G/N . The first isomorphism theorem shows that any surjective homomorphism is (up to isomorphism) a quotient group of the domain.
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