Maybe I should have used the word property instead of "axiom". The quotes are on purpose. I have used "axiom" only because I have seen the term "Group axioms" being used more than "Group properties". And yes the use of "axiom" is a problem. It should be property. But the real problem for me is with the usual definition. Precisely the fact that when the usual definition is translated from natural language to the language of modern mathematics (the language of Set Theory which is a first order language) the translation is not trivial. The last two properties have to be combined together. And for me this is just a Math smell and it should ring a bell that there is indeed a problem.