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Abstract Algebra question?

  1. Oct 25, 2006 #1
    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.

  2. jcsd
  3. Oct 25, 2006 #2
    Do you know what a kernel is?
  4. Oct 25, 2006 #3
    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.
  5. Oct 26, 2006 #4

    matt grime

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    So what is e, in this group, and therefore what is the kernel?
  6. Oct 26, 2006 #5
    e is the identity element.
  7. Oct 26, 2006 #6
    I think he meant, what is the identity elemeen in your group?
  8. Oct 26, 2006 #7
    is 4 the identity ? not sure
  9. Oct 26, 2006 #8
    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"?
  10. Oct 26, 2006 #9


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  11. Oct 26, 2006 #10
  12. Oct 26, 2006 #11


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    Exactly. And now look at your definition of the kernel of Q.
  13. Oct 26, 2006 #12
    so the kernel will be {-1,1}.
  14. Oct 26, 2006 #13


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    Yes, looks good.
  15. Oct 26, 2006 #14
    thanks a lot for helping.
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