Hi. I'm reading a simple introduction to groups. A group is said to be a set satisfying the following axioms (called the 'group axioms'): 1) Associativity. 2) There is a neutral element. 3) Every element has an inverse element. 4) Closure. My questions is simply: why are they called axioms? I thought an axiom was something we take as a starting point, defining it to be true and then deduce something from it (possibly together with other axioms). Why are 1-4 not just the definition of a group?