I am reading Alan Beardon's "Algebra and Geometry" and it says:
(2) for all f, g and h in G, f * (g * h) = (f * g) * h.
(2) is called the associative law, and it says that f * g * h is uniquely defined regardless of which two operations * we choose to do first. The point here is that * only combines two objects at a time, and we have to apply it twice (in some order) to obtain f * g * h. There are exactly two ways to do so and the two must always yield the same results.
The issue I have is that I do not see how is it that there are only two ways to combine two elements out of three. For as I understand I could make six different combinations (if I do not assume commutative property):
(1) ( f * g ) * h
(2) ( f * h ) * g
(3) ( g * f ) * h
(4) ( g * h ) * f
(5) ( h * g ) * f
(6) ( h * f ) * g
And even if I assume that there is a commutative property, that I have not seen stablished in the definition so far, there would still be three different combinations:
(1) ( f * g ) * h
(2) ( f * h ) * g
(3) ( g * h ) * f
Can you please explain to me how is it that there are only two ways to apply a binary operation over three elements and thus defining the associative axiom shall be expressed as f * (g * h) = (f * g) * h and not as ( f * g ) * h = ( f * h ) * g = ( g * h ) * f ?