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

[TrickyDicky]

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

 

[HallsofIvy]

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.

 

[TrickyDicky]

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.

 

[Fredrik]

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 $\phi(G)$.

 

[TrickyDicky]

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.

 

[TrickyDicky]

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."