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Question about a group

  1. Feb 14, 2013 #1
    1. The problem statement, all variables and given/known data
    So we have this operation x*y=x+2y+4
    and then our 2nd one is x*y=x+2y-xy
    I need to check if it is commutative,associative, and if it has a identity and an inverse.
    3. The attempt at a solution
    y*x=y+2x+4 so it is not commutative


    x*(y*z)=x+2(y+2z+4)+4=x+2y+4z+12
    (x*y)*z=(x*y)+2z+4=x+2y+4+2z+4
    Not associative

    Now I will solve for the identity
    e*x=e+2x+4=x
    e=-x-4
    x*e=x+2e+4=x
    e=-2
    Since I have 2 different identity elements this means that one does not exist because
    e should be unique.
    Since there is no e there is no inverse.


    Now for the second one x*y=x+2y-xy
    y*x=y+2x-yx Does not commute

    (x*y)*z=x+2y-xy+2z-xz-2yz+xyz
    x*(y*z)=x+2y+4z-2yz-xy-2xz+xyz
    Not associative

    x*e=x+2e-xe=x
    e=0
    e*x=e+2x-ex=x
    e(1-x)=-x
    The identity element does not seem to be unique so it does not exist.
    Just want to know if I am doing this right.
     
  2. jcsd
  3. Feb 14, 2013 #2

    Mark44

    Staff: Mentor

    What do you mean "our 2nd one"? You can't define the * operation on a group in two different ways.


     
  4. Feb 14, 2013 #3
    The second one is just another problem. They are 2 separate problems.
     
  5. Feb 14, 2013 #4

    Mark44

    Staff: Mentor

    Then you should identify them as such instead of clumping them together as you did.
     
  6. Feb 14, 2013 #5

    Mark44

    Staff: Mentor

    What are you trying to show here? Is the exercise for each to say whether some set with the given operation is a group?

    If so, you don't need to check every group axiom for the operation. For example, if the operation isn't associative, then that's enough to say that the set and the operation aren't a group.

    Also, just because something isn't unique, that doesn't mean it doesn't exist.
     
  7. Feb 15, 2013 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    More fundamentally, the identity can't depend upon "x".

     
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