I was checking multiplication tables for groups with 4 elements, to see which tables "passed" the group axioms of closure, associativity, identity and inverses. But then I had a question, so hopefully someone can help me with this basic group theory question.
The proof that inverses are unique in a group depends upon the associativity axiom. Let a be an element that has two inverses b and c. Then b = be = b(ac) = (ba)c = ec = c. Thus the inverse is unique since b must equal c.
So my question is this. If we see a multiplication table for a finite group, and we can easily check closure, existence of identity, and existence of inverses, and further we can see that inverses are all unique, does this necessarily imply that the last axiom, associativity, holds? If yes, how?