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Abstract Algebra.

  1. Feb 6, 2007 #1
    Please I need your help for that qustion and how do slove that qustion's problem. can you help me for slove for that? Pleasee

    Let G be any group, and let a, b ∈ G. Show that there are c, d ∈ G such that ac = b and da = b. Hint: you have to give an explicit definition for c and d in terms of a and b.
     
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  3. Feb 6, 2007 #2

    mathwonk

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    what have you tried?
     
  4. Feb 6, 2007 #3
    I have tried set up matrices for that but not work. I don't know how to slove for this problem...


    Let G be any group, and let a, b ∈ G. Show that there are c, d ∈ G such that ac = b and da = b. Hint: you have to give an explicit definition for c and d in terms of a and b.
     
  5. Feb 6, 2007 #4

    morphism

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    Matrices?? All you need is:
    (i) if x is in G, then so is its inverse
    (ii) if x,y are in G, then so is xy
     
  6. Feb 6, 2007 #5
    i m tried work on this..

    GIVE a, b ∈ G
    SHOW c, d ∈ G ???

    ac=b da=b

    ac=b ---> proof: a (a^(-1)b)=b (IS THAT RIGHT? I THINK SO AND THAT'S RIGHT)

    da=b ---> proof: ???
     
  7. Feb 6, 2007 #6
    if i want to say about ac=b

    proof: a (a^(-1)b)=b
    uniqueness
    if ac=ad=b
    then c=d (left Cancellation)

    How do I slove for da=b?? I need for that..
     
  8. Feb 6, 2007 #7

    morphism

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    Same idea: let d = ba^(-1).
     
  9. Feb 8, 2007 #8
    Same idea: let d = ba^(-1). ?? I seem that not enough

    ac=b
    a (a^(-1)b)=b, if ac=ad=b, then c=d (left cancellation).. I m sure that's right.



    da=b
    d=ba(a^(-1)), if ad=ac=b, then d=c (right cancellation) is that right?????
     
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