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Prove the following is a group

  1. Feb 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that a nonempty set S with an associative binary operation * on S such that a*c=b and d*a=b have solutions in S for all a,b in S is a group.


    2. Relevant equations



    3. The attempt at a solution
    In order to prove something is a group, I know that I must show associativity of *, existence of identity element and inverse for each element in S.
    - We are given that S has an associative binary operation, so that is satisfied.
    - In regards to finding the identity element such that a*e=e*a=a for all a in S...this is where I am not sure where to begin. Any suggestions on how to start?
     
  2. jcsd
  3. Feb 15, 2009 #2

    Dick

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    Take a and b to be the same element a. Then your premise tells you you can find c and d such that a*c=a and d*a=a. It looks like c and d should be your identity 'e', right? To show that's true, you have to prove c=d, and you have to prove if you pick another element besides a, you get the same identity. It can be done. You just have to keep using the only premise you've got over and over. Similar course for inverses.
     
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