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Klein 4-group 2 Associative property

  1. Dec 8, 2013 #1
    1. The problem statement, all variables and given/known data
    How to prove in the easiest way that Klein 4-group is associative.

    2. Relevant equations
    Four elements ##a^2=b^2=c^2=e^2=e##.


    3. The attempt at a solution
    If that is group with four elements, how many types of
    ##a*(b*c)=(a*b)*c## I need to have?
    1) ##e*(e*e)=e*e=e##
    ##(e*e)*e=e*e=e##
    2) ## e*(a*a)=e*e=e##
    ##(e*a)*a=a*a=e##
    3) ## e*(b*b)=e*e=e##
    ##(e*b)*b=b*b=e##
    3) ## e*(c*c)=e*e=e##
    ##(e*c)*c=c*c=e##
    4) ##e*(e*a)=e*a=a##
    ##(e*e)*a=e*a=a##
    5) ##e*(e*b)=e*b=b##
    ##(e*e)*b=e*b=b##
    6) ##e*(e*c)=e*c=c##
    ##(e*e)*c=e*c=c##
    7) ##a*(a*a)=a*e=a##
    ##(a*a)*a=e*a=a##
    8)##b*(b*b)=b*e=b##
    ##(b*b)*b=e*b=b##
    9) ##c*(c*c)=c*e=c##
    ##(c*c)*c=e*c=c##
    ...
    how many of this do I have? Could you tell me the number of this. For example 20) or something? What is the easiest way to prove associativity?
     
    Last edited: Dec 8, 2013
  2. jcsd
  3. Dec 10, 2013 #2
    Any idea?
     
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