Why are the group axioms called "axioms"? My book says:

A group is an ordered pair $(G, \cdot)$ where $G$ is a set and $\cdot$ is a binary operation satisfying the following axioms [emphasis mine]:
$\vdots$

Why doesn't it say

A group is an ordered pair $(G, \cdot)$ where $G$ is a set and $\cdot$ is a binary operation satisfying the following propreties:
$\vdots$

As far as I know, axioms are statements which we take for granted; I don't understand why the group axioms are something we have to take for granted; we already know that there exist objects (such as the set of permutations of $3$ objects) which satisfy those propreties.
Even if we knew of no objects which satisfy those propreties, then perhaps the axiom should be "There exists an object which satisfies those propreties" rather the axioms being the propreties themselves.

Quoting from Wikipedia: “An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments”. Don't you agree that group axioms are starting points?
