# Is the “binary operation” in the definition of a group always deterministic?

The introduction to group theory that I'm reading requires that the actions of a group are "deterministic"; but the formal definition given makes no mention of this property:

A set G is a group if the following criteria are satisfied.

1. There is a binary operation $\cdot$ on $G$.
2. That operation is associative...
3. There is an identity element $e$...
4. Every element in $g\in G$ has an inverse...

Neither do other definitions I'm familiar with (which only mention the operation and the axioms of closure, associativity, identity element, and inverse element).

Where did this property go? Is it really a property of groups or only of certain kinds of groups? Does it follow from other properties or axioms?

I'm guessing that another way to express "deterministic" is that for $a,b,c\in G$, $a=c \land b=c \implies a=b$, but I don't see that in the definition either.

• It simply means that the operation is a function from $G\times G$ to $G$: the value of $a\cdot b$ is completely determined by $a$ and $b$. – Brian M. Scott May 23 '12 at 21:06
• More generally, functions —as far as this word is understood in common mathematical use— are "deterministic".(Your last remark is has not much to do with determinism: it is simply the statement that equality is a tranitive relation) – Mariano Suárez-Álvarez May 23 '12 at 21:12
• Moreover the logical formula in your last paragraph is implied by the axioms of equality. – Zhen Lin May 23 '12 at 21:13
• @MarianoSuárez-Alvarez: Isn't my "guess" saying the same thing as Brian's comment? That $x \cdot y$ must always yield the same result, so that if $c \equiv x \cdot y = a$ and $c \equiv x \cdot y = b$, then $a$ and $b$ must be the same? (Remember, I'm a beginner.) – orome May 23 '12 at 21:30
• Yes, that uniqueness of the result would be part of the definition of "operation". – GEdgar May 23 '12 at 21:50

Please note, that I refer here to theory, not practice and nature, I have no idea how The_Real_World$^{(TM)}$ looks on the inside ;-)
For example, take the aforementioned random variables. If you dare to go deep enough, you will find that they are, curiously, deterministic! Every random variable is a function $X : \Omega \to \mathbb{R}$ (or some other set of values of your choice) and nothing more. You can interpret it in a way that it is "random", however, deep in guts, it is not, just fix some $\omega \in \Omega$ and that's it, nothing moves, nothing changes. Please note, that this is a big strength, to tame the wild nature into plain, repeatable experiments and deterministic formulas.