1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Homomorphisms and kernals

  1. Oct 27, 2007 #1
    1. The problem statement, all variables and given/known data

    Let R* be the group of nonzero real numbersunder multiplications. Then the determinant mapping A->det A is a homomorphism from GL(2,R) to R* . The kernel of the determinant mapping is SL(2,R).
    2. Relevant equations

    3. The attempt at a solution

    I know det(A)det(B)=det(AB) but other than knowing that property, I don't understand the meaning of the kernel nor SL(2,R) nor do I understand how GL(2,R) is a homomorphism. I know SL(2,R) stands for Special linear group and GL(2,R) General Linear group.
  2. jcsd
  3. Oct 28, 2007 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    GL(2,R) isn't a homomorphism. It is a group. Your post clearly states that the determinant map is a homomorphism, not GL(n,R).

    What is det(I), I the identity?

    Doesn't this show that det satisfies the definition of homomorphism?

    You do understand what SL(2,R) is - you wrote out its definition: the set of matrices of determinant 1.

    The kernel is the set of matrices sent to the identity...
  4. Oct 28, 2007 #3
    Isn't the kernel the set of stuff that is sent to 0, not the identity?
  5. Oct 28, 2007 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Groups aren't even required to have an element called '0'!
  6. Nov 13, 2007 #5
    Yeah true, so the definition I had must've been for something with identity=0. So is it really what is sent to the identity?
  7. Nov 13, 2007 #6
    In a group, the identity is often denoted e. The zero is the additive identity element in a ring. The wikipedia article calls the identity element (in a group) 1.

    So, calling the additive identity either 0 or 1 is generally a bad idea. (the '1' is actually 0 in the group of integers and all of its subgroups, and calling it '0' is...a ring thing)

    The kernel of a group homomorphism phi:A->B is the preimage of {e_B} under phi, e_B the identity element in e_B. The preimage of a subset S of B under a function f:A->B is defined set theoretically as {x in A : f(x) in S}.

    So for your problem: What's the identity element in R*? (Certainly not zero!) What's the preimage of this identity under the group homomorphism given by the determinant?

    These should all be obvious to you.

    (Note: SL(n,R) is defined as the nxn matrices over R with determinant 1. Exercise: Show that this is a subgroup of GL(n,R))
    Last edited: Nov 13, 2007
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook