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
Let’s say you have a function f:G->H, ie it maps elements of group G to elements of group H. So:
1. “Onto”: every element in H is mapped by f, ie for every h in H there’s some g in G such that f(g)=h. This definition applies to all functions, not just homomorphisms.
2. “one-to-one”: any two different elements in G are mapped to different elements in H, ie if f(x)=f(y) then x=y. As above, this applies to all functions.
3. Monomorphism is another name for a one-to-one homomorphism. It also applies to other fields in abstract algebra, not just group theory.
4. Isomorphism is a himimorphism that is both one-to-one and onto. This means that each element in G is uniquely mapped to an element in H and vice versa. This makes G and H “equivalent”, as if they are the same group but with names for their elements.
5. Yes it’s also endomorphism. It’s not an automorophism because it’s not onto.
6. Endomorphism is a homomorphism from a group to itself.
7. Automorophism is an isomorphism between a group and itself (ie an endomorphism that is also one-to-one and onto).
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