# Abstract Algebra question?

1. Oct 25, 2006

### 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

2. Oct 25, 2006

### gonzo

Do you know what a kernel is?

3. Oct 25, 2006

### raj123

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.

4. Oct 26, 2006

### matt grime

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

5. Oct 26, 2006

### raj123

e is the identity element.

6. Oct 26, 2006

### gonzo

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

7. Oct 26, 2006

### raj123

is 4 the identity ? not sure

8. Oct 26, 2006

### gonzo

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"?

9. Oct 26, 2006

10. Oct 26, 2006

### raj123

11. Oct 26, 2006

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

12. Oct 26, 2006

### raj123

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

13. Oct 26, 2006