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Unique Solutions in a Group

  1. Oct 12, 2008 #1
    1. The problem statement, all variables and given/known data

    let a,b [tex]\in[/tex]G show that ax=b has a unique solution in G

    3. The attempt at a solution

    i know what needs to be done, i just dont know how to do it.

    Want to prove:

    1. There is a solution
    2. solution is unique

    to prove uniquness of a soltuion just suppose you have a different solution x' and show that x'=x

    to show that there is a solution (im not sure this part is right, cause it seems too simple) simply multiply (left) by [tex]a^{-1}[/tex]

    that is [tex]a^{-1}[/tex]*ax=[tex]a^{-1}[/tex]b

    we can do this because a is in the group, which implies [tex]a^{-1}[/tex] is in the group

    THis is where i am stuck... i really dont know how to proceed. Have i even done the first part right?

    any help appreciated
  2. jcsd
  3. Oct 12, 2008 #2


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    Science Advisor
    Homework Helper

    So you showed that there is a solution, namely [itex]x=a^{-1}b[/itex]. To show that x is unique, use the fact that in a group, inverses are unique.
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