A question about groups: may I substitute a binary operation with a function? I have a fundamental question about groups. Consider the definition from Wolfram Mathematica:

A group  is a finite or infinite set of elements together with a binary operation (called the group operation) that together satisfy the four fundamental properties of closure, associativity, the identity property, and the inverse property. 

In this definition, may I substitute a binary operation with a function, say something like $f(a,b)$ where $f$ is not necessarily a simple operator like addition?
 A: I guess the biggest reason for writing $a+b / ab$ is the associativity law looks uncluttered.
Here's function notation for associativity: $f(a,f(b,c)) = f(f(a,b),c)$.
Here's 'normal' notation $(ab)c = a(bc)$ - which one looks better?
A: A couple of points. First, you write "may I substitute a binary operation with a function". Note that a binary operation is a function, satisfying some extra properties. And there are plenty of contexts where it isn't something as simple as addition. One of the first examples of abstract groups that you meet are permutation groups, where the operation is composition of permutations. This is indeed just one of many examples where the elements of the group are functions of some kind (in this case, a permutation is just a bijection on some finite set). Since function composition is rarely abelian, this is one way that non-abelian structure arises "in nature". If you want to be cute about it, by Cayley's theorem that all finite groups embed into a permutation group, all group laws on finite groups can be thought of as function composition.
