# 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. – Dylan Moreland Jan 7 '12 at 2:27
• Is your question really about groups, or are you actually asking 'What can be a binary operation?' – matt Jan 7 '12 at 2:30
• My question is about binary operation in the context of group theory. – Ravi Kulkarni Jan 7 '12 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. – Srivatsan Jan 7 '12 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 – bradhd Jan 7 '12 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. – Joel Cohen Jan 7 '12 at 17:56