Possible title: When Does the Kernel of a Homomorphism Reduce to the Identity?

In summary, the kernel of a group homomorphism may reduce to the identity under certain circumstances."
  • #1
TrickyDicky
3,507
27
I'm centering on lie group homomorphisms that are also covering maps from the universal covering group. So that if their kernel was just the identity
they would be isomorphisms.
Are there situations in which the kernel of such a homomorphism would reduce to the identity? I'm thinking of situations where the groups act on different sp
 
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  • #2
I'm not clear what your question is. You state that you know that if a homomorphism is an isomorphism then its kernel is the identity. You then ask "Are there situations in which the kernel of such a homomorphism would reduce to the identity?" Do you mean "are the situations in which the kernel is the identity but the homomorphism is NOT an isomorphism?

If that is your question, then, strictly speaking, yes. In order to be an isomorphism, a homomorphism must be both "one to one" and "onto". A homomorphism is "one to one" if and only if its kernel is the identity. But that leaves "onto". It is possible for a homomorphism to be "one to one" yet not be "onto".

However, in such a case, the homomorphism is "onto" a subgroup of the original range group and we normally then focus on the homomorphism being an isomorphism to that subgroup.
 
  • #3
HallsofIvy said:
I'm not clear what your question is. You state that you know that if a homomorphism is an isomorphism then its kernel is the identity. You then ask "Are there situations in which the kernel of such a homomorphism would reduce to the identity?" Do you mean "are the situations in which the kernel is the identity but the homomorphism is NOT an isomorphism?

If that is your question, then, strictly speaking, yes. In order to be an isomorphism, a homomorphism must be both "one to one" and "onto". A homomorphism is "one to one" if and only if its kernel is the identity. But that leaves "onto". It is possible for a homomorphism to be "one to one" yet not be "onto".

However, in such a case, the homomorphism is "onto" a subgroup of the original range group and we normally then focus on the homomorphism being an isomorphism to that subgroup.
Hi, that was not what I meant, I had some problem with the editor and the question appeared cut.
I was going to give some example: consider the well known homomorphism SU(2)->SO(3), Since SU(2) is the Universal cover of SO(3) the only reason it is not an isomorphism is that its kernel has one more element besides the identity, -I, and my question is if there are situations where the kernel
of this homomorphism is reduced to the identity, amd therefore turned into an isomorphism. Like maybe wnen acting on spaces that are not vector spaces that habe isometries that reduce the kernel.
 
  • #4
I'm not sure what you mean by "situations". Are you asking about group homomorphisms ##\phi:G\to H## where G and H are not necessarily SU(2) and SO(3), or are you asking specifically about SU(2) and SO(3)?

I suspect that whatever the exact question is, the best answer you will get is that there's a theorem that says that if ##\phi:G\to H## is a homomorphism, then the quotient group ##G/\ker\phi## is isomorphic to ##\phi(G)##.
 
  • #5
Fredrik said:
I'm not sure what you mean by "situations". Are you asking about group homomorphisms ##\phi:G\to H## where G and H are not necessarily SU(2) and SO(3), or are you asking specifically about SU(2) and SO(3)?

I suspect that whatever the exact question is, the best answer you will get is that there's a theorem that says that if ##\phi:G\to H## is a homomorphism, then the quotient group ##G/\ker\phi## is isomorphic to ##\phi(G)##.
/
That was just an example that fulfills the conditions I stablished. By situations I mean there are group actions on manifolds other than the usual Rn that modify the kernel so that the quotient trivially becomes G/Identity.
 
  • #6
Basically I guess I was trying to define a non-canonical or unnatural isomorphism, that is, an isomorphism that requires a previous choice, whereas canonical means: "distinguished representative of a class", particularly one that does not require making any choice; this is also known as "natural", as in natural transformation."
 

1. What is a kernel of homomorphisms?

The kernel of homomorphisms is a mathematical concept that refers to the set of elements in the domain of a homomorphism that map to the identity element in the codomain. In other words, it is the set of elements that are mapped to the neutral element of a group or algebraic structure by a homomorphism.

2. How is the kernel of homomorphisms calculated?

The kernel of homomorphisms can be calculated by finding the elements in the domain that are mapped to the identity element in the codomain. This can be done by applying the homomorphism to each element in the domain and checking if the resulting element is the identity element.

3. What is the significance of the kernel of homomorphisms?

The kernel of homomorphisms is important because it provides information about the structure of the domain and codomain of a homomorphism. It can also be used to determine the injectivity and surjectivity of a homomorphism and to define the quotient structure of a group or algebraic structure.

4. How does the kernel of homomorphisms relate to the image of a homomorphism?

The kernel of homomorphisms and the image of a homomorphism are closely related. The image is the set of elements in the codomain that are mapped by the homomorphism, while the kernel is the set of elements in the domain that are mapped to the identity element. These two sets are complementary and their union is equal to the entire domain.

5. Can the kernel of homomorphisms be empty?

Yes, the kernel of homomorphisms can be empty if the homomorphism is injective. In this case, there are no elements in the domain that are mapped to the identity element in the codomain, so the kernel is empty. This is often the case for isomorphisms, as they preserve the structure of the domain and do not map any elements to the identity element.

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