Klein 4-group 2 Associative property

[LagrangeEuler]

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?

 

[LagrangeEuler]

Any idea?