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Group theory question

  1. Jan 6, 2015 #1
    1. The problem statement, all variables and given/known data
    Let p: G-->M be a group homomorphism with ker(p) = K. If a is an element of G, how that Ka = {g in G | p(g) = p(a)}

    2. Relevant equations
    none needed

    3. The attempt at a solution
    Okay, I've been struggling with this problem for awhile and I've ran into a problem:

    -Let g be an element of Ka
    -Let b be an element of K such that ba = g.

    Since g is an element of Ka and the intersection of Ka and K is {1}, p(g) does not equal zero.

    But then if ba = g then:
    p(ba) = p(g)
    p(b)p(a) = p(g)
    0p(a) = 0, but p(g) can't be zero!!

    Someone wanna shed some light perhaps? I guess I need help on understanding this road block I've run into as well as the actual problem >.<. Thanks!
     
  2. jcsd
  3. Jan 6, 2015 #2

    vela

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    Why do you say p(b)=0?
     
  4. Jan 6, 2015 #3
    Nevermind. Lol i'm a noob i had an error in how I was thinking.
     
  5. Jan 6, 2015 #4

    pasmith

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    What is zero? You're dealing with groups. 1 is the identity and all elements are invertible.
     
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