TITLE: A question about groups: may I substitute a binary operation with a function?
QUESTION [6 upvotes]: 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?

REPLY [6 votes]: 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?