- #1
LagrangeEuler
- 717
- 20
Homework Statement
How to prove in the easiest way that Klein 4-group is associative.
Homework Equations
Four elements ##a^2=b^2=c^2=e^2=e##.
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?
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