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).
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.
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).
First, think of a function as a way to turn one set into another set. If I have a set A={a,b,c} and you have a set B={x,y}, then a function telling us where everything in A goes can be thought of as turning A into B. You can choose a lot of different functions for any two sets because you don’t have to worry about operations.
Once you equip your sets with operations to create groups, you may want to be more careful about which functions you use. Generally speaking, a homomorphism is a way to turn one group into another group while preserving the operation. This means that when you operate two things together in the first group, then send them to the second group, it better be exactly the same as first sending them into the second group, and then operating them over there instead!
In more detail, suppose group G has operation *, so you can combine two elements in G by doing a*b. Then you have group H with it’s operation being +, so you can combine two elements in H by doing x+y.
Now say you want to send elements of G into H, so given an element “a” living in G, you want to send it over H be using a function. If the function is named f, then a lives in G, while f(a) lives in H. Given a second element in G, called “b,” you can also send it over to H where it turns into f(b).
Since f(a) and f(b) are now living in H you can put them together with the operation of H. In other words, you can calculate f(a)+f(b). However, we could go back and calculate a*b back in G. Whatever the hell that happens to be, it’s still something in G, so we can send it over to H via the function f. So, a*b lives in G, while f(a*b) lives in H.
This creates a possible issue. We have two things living in H:
f(a*b) and f(a)+f(b).
Which really feel like the same thing, but we just took two different paths to get there. Either operate the elements then function the result, or function both elements then operate them afterwards.
Therefore we really want functions that have:
f(a*b) = f(a)+f(b)
for any choice of a,b. This is the sense in which you can properly turn not only the elements of G into the elements of H, but you can do so while preserving the operation of G.
Once you understand a homomorphism properly the rest of your questions are easier to answer.
An onto function is one where everything in the second set is targeted by the function. There are no elements in the second set that are not “hit” by something in the first set. So if I have a set A={a,b,c} and you have a set B={x,y}, a function defined by:
f(a)=x, f(b)=x, f(c)=y
would be onto because both x and y are hit by something.
Meanwhile a function defined by:
f(a)=x, f(b)=x, f(c)=x
would not be onto because not everything in the second set was hit. A homomorpshim, being a function, can also be onto, as an additional nice property.
A one to one function is dual to an onto function. Instead of needing everything in the second set to be hit, an onto function is one where whatever is hit, is only hit once. So if I have a set A={a,b,c} and you have a set B={x,y,z,w} then a function defined by:
f(a)=x, f(b)=y, f(c)= w
would be one to one because x, y, and w were each hit only once. z was not hit at all, but that’s fine, that just means the function is not onto, but it is still one to one.
A monomorphism is a homomorphism that is also one to one.
An isomorphism is a homomorphism that is both one to one and onto. When this occurs, this means that the two groups are actually the “same” group up to a renaming of the elements. A lot more can be said about this, but I’m getting tired!
An endomorphism is a homomorphism where the same group is both the domain and target of the function, while an automorphism is an isomorphism of this kind of function.
First, think of a function as a way to turn one set into another set. If I have a set A={a,b,c} and you have a set B={x,y}, then a function telling us where everything in A goes can be thought of as turning A into B. You can choose a lot of different functions for any two sets because you don’t have to worry about operations.
Once you equip your sets with operations to create groups, you may want to be more careful about which functions you use. Generally speaking, a homomorphism is a way to turn one group into another group while preserving the operation. This means that when you operate two things together in the first group, then send them to the second group, it better be exactly the same as first sending them into the second group, and then operating them over there instead!
In more detail, suppose group G has operation *, so you can combine two elements in G by doing a*b. Then you have group H with it’s operation being +, so you can combine two elements in H by doing x+y.
Now say you want to send elements of G into H, so given an element “a” living in G, you want to send it over H be using a function. If the function is named f, then a lives in G, while f(a) lives in H. Given a second element in G, called “b,” you can also send it over to H where it turns into f(b).
Since f(a) and f(b) are now living in H you can put them together with the operation of H. In other words, you can calculate f(a)+f(b).
However, we could go back and calculate a*b back in G. Whatever the hell that happens to be, it’s still something in G, so we can send it over to H via the function f. So, a*b lives in G, while f(a*b) lives in H.
This creates a possible issue. We have two things living in H:
f(a*b) and f(a)+f(b).
Which really feel like the same thing, but we just took two different paths to get there. Either operate the elements then function the result, or function both elements then operate them afterwards.
Therefore we really want functions that have:
f(a*b) = f(a)+f(b)
for any choice of a,b. This is the sense in which you can properly turn not only the elements of G into the elements of H, but you can do so while preserving the operation of G.
Once you understand a homomorphism properly the rest of your questions are easier to answer.
An onto function is one where everything in the second set is targeted by the function. There are no elements in the second set that are not “hit” by something in the first set. So if I have a set A={a,b,c} and you have a set B={x,y}, a function defined by:
f(a)=x, f(b)=x, f(c)=y
would be onto because both x and y are hit by something.
Meanwhile a function defined by:
f(a)=x, f(b)=x, f(c)=x
would not be onto because not everything in the second set was hit.
A homomorpshim, being a function, can also be onto, as an additional nice property.
A one to one function is dual to an onto function. Instead of needing everything in the second set to be hit, an onto function is one where whatever is hit, is only hit once. So if I have a set A={a,b,c} and you have a set B={x,y,z,w} then a function defined by:
f(a)=x, f(b)=y, f(c)= w
would be one to one because x, y, and w were each hit only once. z was not hit at all, but that’s fine, that just means the function is not onto, but it is still one to one.
A monomorphism is a homomorphism that is also one to one.
An isomorphism is a homomorphism that is both one to one and onto. When this occurs, this means that the two groups are actually the “same” group up to a renaming of the elements. A lot more can be said about this, but I’m getting tired!
An endomorphism is a homomorphism where the same group is both the domain and target of the function, while an automorphism is an isomorphism of this kind of function.
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