An alternative formulation of this question is: why is the set of four axioms that define a group so fundamental, rather than, for example, the same four axioms except that associativity is interchanged with commutativity?
What makes this particular set of 4 axioms so special that abstract algebra is based on it?
The four axioms for a group $(G, +)$ are:
- Closure: $\;\forall a, b \in G$, $\;(a+b)\in G$
- Associativity: For all $a, b, c \in G,\,$ $\;a+(b+c)=(a+b)+c$.
- Identity: $\;\exists e \in G$ such that $\forall a \in G,\; a+e=e+a = a$.
- Invertibility: $\forall a \in G, \exists b = -a$ such that $b+a=a+b= e$.
If we add commutativity we have a commutative group. Why not for example make commutativity the key axiom and speak of an "associative group" as a special case? Or why not something else entirely?
What's so special about this combination of 4 axioms?
Edit: For example, is there a theory of "commutative non-associative magma's with an identity and inverses"?