Here is what I’ve been doing for the past hour: proving that associativity does not follow from other group axioms including that the left and right inverses are the same and the neutral element is unique.

Let us define the operation * on R, the reals as follows. For each x,y in R let x*y=x+y if x = – y or |x+y| > 1, otherwise define x*y=x if |x|>|y| and x*y=y if |y|>|x|. Now (R,*) has a neutral element, it is 0. Every element has an inverse and it is even commutative, but associativity fails: take x=y=1/4 and z=1. Then (x*y)*z=1/4*z=1.25, but x*(y*z)=1/4*(1.25)=1.5.

Exercise: invent a less ad-hoc example.

Neljän alkion laskutoimitustaulu:

a b c e

——————–

a | a e b a

b | e b a b

c | b a e c

e | a b c e

-tulo pysyy joukossa

-e on neutraalialkio

-a ja b toistensa käänteisalkioita, e ja c omia käänteisalkioitaan

Kuitenkin:

(a+b)+c=e+c=c

a+(b+c)=a+a=a