I totally agree that those are properties and not axioms. This is why I put them in quotations. The reason why I did this is because usually they are called Group axioms or so I have most seen them being called.

I agree we do no't have do ask the identity to be unique it follows from the associativity of operation and it's existence. Actually in non of the definitions I put in mine blog post I have required/stated that it should be unique, even in the last one.

Yes I totally agree that if there isn't an identity the thing is not a Group and it's end of story. :)

But I strongly disagree with your point that if there exists an identity only then we should check for an inverse. And this is exactly why I say that this definition is smelly. Because the properties we want/require should be independent of one another (it should not matter in which order we check them). Also non of them should be based on a collary from one or more of the others. And it should be straightforward if the definition is given in natural language to be translated in a formal and logically correct one.

And as I showed the usual definition with or without the remark (it does not really matter) requires an non-trivial translation. The second and the third properties have to be merged in one. I believe this should ring a bell that there is something not OK with the definition.