Here, I (basically) stated the group axioms as follows.
- $(xy)z=x(yz)$
- $xe=x, ex=x$
- $xx^{-1}=e$
In that post, answerers Martin and Ittay were critical of the above list for not including $x^{-1}x=e$, even though it follows from the above three. Pece's answer also included $x^{-1}x=e$ without comment.
How can I tell whether a system of axioms of 'complete'? Is this even a rigorous concept?
even if we delete ex=x, it is still a group. which follows like this:
$$ x^{-1}x=x^{-1}xe=x^{-1}x(x^{-1}(x^{-1})^{-1})=x^{-1}(xx^{-1})(x^{-1})^{-1}=x^{-1}(e)(x^{-1})^{-1}=x^{-1}(x^{-1})^{-1}=e $$
and
$$ ex=xx^{-1}x=x(x^{-1}x)=xe $$
so it is a group.
I am not sure if there is a general way to tell a group, but the basic rule of a group are those four: closure, associativity, identity and inverse.
As long as we can conclude the four rules from what we have given, then it is complete. So I think concentrate on the basic idea of the system is important, all others come from the basics.
As for group, we often consider subgroup, which will simplify the judging process, we just consider $xy^{-1}$ belongs to the subset or not.