# 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 binary operation on a set $S$ is a function $f\colon S \times S \to S$. We write $ab$ or $a + b$ (the latter is usually reserved for abelian groups) instead of $f(a, b)$ because it's shorter, but even addition on the integers can be thought of as a function in this way. Commented Jan 7, 2012 at 2:27
• Is your question really about groups, or are you actually asking 'What can be a binary operation?'
– matt
Commented Jan 7, 2012 at 2:30
• My question is about binary operation in the context of group theory. Commented Jan 7, 2012 at 2:32
• Sometimes functions arising in special contexts are given specific names; apart from the vanilla function, one typically comes across terms like operation, operator, map, transform, and transformation. Commented Jan 7, 2012 at 2:35
• To be pedantic, the term "algebraic group" generally refers to a particular kind of group, as laid out in the Wikipedia article en.wikipedia.org/wiki/Algebraic_group Commented Jan 7, 2012 at 3:33

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?
• I think the notation $a+b$ et $ab$ are very old, so I'm guessing things happened the other way around : the notation $a+b$ and $ab$ existed for a very long time, and at some point (when came abstract algebra) we realized associativity was what made the 'binary operator' notation more practical that function one. Commented Jan 7, 2012 at 17:56