From the way content is written online I am having trouble discerning which properties of numbers require proof and which are taken to be axioms because so few take the time to explain anything from a more fundamental level.
What I see most often for an arbitrarily defined group (just picking a random letter) S containing a set (not necessarily with order) of natural numbers is that:
-integers are closed under addition
-integers are associative
-there exists the identity element
-that there is an inverse element
Now I could have sworn at some point that I was asked at least to prove the existence of an identity element within a set of natural numbers a long time ago in precalculus, but because these statements are bunched together so often, I have to ask: are these statements taken to be self-evident among mathematicians or must they be constructed from a more fundamental level?