I’m taking number theory online right now and we are starting with group theory. I’ve tried self studying before but I just cannot grasp how to read the language.
Could somebody please explain the following in simple terms:
1. Onto (in terms of group homomorphisms)
2. One-to-one (in terms of group homomorphisms)
3. Monomorphism
4. Isomorphism
5. How is f(x)=2x a monomorphism from Z =(Z,+) to Z=(Z,+)? Wouldn’t this be an endomorphism or an automorphism since it is mapping Z to Z?
6. Endomorphism
7. Automorphism
In: 1
Remember that group homomorphism is a function from a group to a group with a particular property. So group homomorphism is also functions. Knowing that should make a lot of terminology easier.
“Onto”, “surjective” and “epimorphism” means the same thing in the context of group homomorphism. And it means the function is onto/surjective (in the usual sense of function).
“One-to-one”, “injective”, and “monomorphism” means the same thing in the context of group homomorphism. And it means the function is one-to-one/injective (in the usual sense of function).
Isomorphism means both onto and one-to-one. This means the function is bijective (in the usual sense of function).
Endomorphism means the function go from the group to itself. Being an “endomorphism” and being a “monomorphism” are *not* mutually exclusive property, because the 2 terms refer to unrelated properties: the first term tells you the codomain of the function, the second term tells you where the elements of the function must go.
Automorphism means both endomorphism and isomorphism.
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