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
A group homomorphism is a kind of function that operates on elements of a group, and some of these terms are just general terms for functions.
Some vocabulary: *domain* is the set of input values, *codomain* is the set from which output values are taken, *range* is the set of actual output values. If that’s unclear, think of the function y=x²: ℝ⇒ℝ. The domain and codomain are both ℝ, but the range is [0, ∞). Now to answer your questions:
1. A function is *onto* (AKA *surjective*) when the range equals the codomain.
2. A function f is *one-to-one* (AKA *injective*) when, for every x and y in the domain, if f(x) = f(y) then x=y.
3. A *monomorphism* is a one-to-one group homomorphism.
4. An *isomorphism* is a bijective group homomorphism, i.e. one that is both one-to-one and onto.
5. I’ll come back to this one once the other terms you ask about later are defined.
6. An *endomorphism* is a group homomorphism where the domain equals the codomain.
7. An *automorphism* is a bijective endomorphism.
Now back to #5. These definitions are not mutually exclusive; a single function can fit any number of them. The function you gave fits the definition of monomorphism because it is one-to-one and fits the definition of “group homomorphism” (which you didn’t ask about so I assume you know). Every distinct output value is the result of the function being applied to a distinct input value. It’s also an endomorphism, because both the domain and codomain are ℤ with the operation being addition. However, it’s not an automorphism because it’s not onto (no odd numbers in the codomain have corresponding values in the domain).
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