Examples of a cayley table that represents a structure that satisfies all group axioms except associativity I'm curious if there are any cayley tables on a finite amount of elements that satisfy the axioms of a) closure, b) identity, and c) inverse, but that for at least one triple of elements do not satisfy the associative property, and so thus the set is not a group.
I'm also wondering if there are any in which every element has a unique inverse but a group is still not formed because associativity is not held.

For the table
$$\matrix{e&a&b\cr a&e&e\cr b&b&e\cr}$$
we have
$$(ab)a=ea=a\quad\hbox{but}\quad a(ba)=ab=e\ .$$
Another example:
$$\matrix{e&a&b\cr a&e&a\cr b&b&e\cr}\ ,\qquad (ab)a=e\ ,\qquad a(ba)=a\ .$$
