Proving Homomorphism and Finding Kernel in Abstract Algebra

[raj123]

Given R=all non-zero real numbers.

I have a mapping Q: R-> R defined by Q(a) = a^4 for a in R. I have to show that Q is a homomorphism from (R, .) to itself and then find kernel of Q.

In order to prove homomorphism i did this, for all a, b in R
Q(ab) = (ab)^4 = a^4b^4 = Q(a)Q(b).

Is this correct way? Also how do i find the kernel of Q.

thanks

 

[gonzo]

Do you know what a kernel is?

 

[raj123]

Do you know what a kernel is?

if O:G -> H is a homomorphism , then the kernel of O is the set of all elements a in G such that O(a) = e of H(identity of H).The kernel of a homomorphism is always a subgroup of the domain.

 

[matt grime]

So what is e, in this group, and therefore what is the kernel?

 

[raj123]

So what is e, in this group, and therefore what is the kernel?

e is the identity element.

 

[gonzo]

e is the identity element.

I think he meant, what is the identity elemeen in your group?

 

[raj123]

I think he meant, what is the identity elemeen in your group?

is 4 the identity ? not sure

 

[gonzo]

is 4 the identity ? not sure

It seems a more important question for you is do you know what an identity element is at all? Do you understand what is meant by "identity element"?

 

[radou]

is 4 the identity ? not sure

Look at your group (R, .) and read the definition: http://mathworld.wolfram.com/IdentityElement.html" .

 

[raj123]

Look at your group (R, .) and read the definition: http://mathworld.wolfram.com/IdentityElement.html" .

so 1 is the identity.

 

[radou]

so 1 is the identity.

Exactly. And now look at your definition of the kernel of Q.

 

[raj123]

Exactly. And now look at your definition of the kernel of Q.

so the kernel will be {-1,1}.

 

[radou]

so the kernel will be {-1,1}.

Yes, looks good.

 

[raj123]

Yes, looks good.

thanks a lot for helping.