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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?

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    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.2012-01-07
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    Is your question really about groups, or are you actually asking 'What can be a binary operation?'2012-01-07
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    My question is about binary operation in the context of group theory.2012-01-07
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    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*.2012-01-07
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    To be pedantic, the term "algebraic group" generally refers to a particular kind of group, as laid out in the Wikipedia article http://en.wikipedia.org/wiki/Algebraic_group2012-01-07

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