# 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.

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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-Alvarez 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.) – raxacoricofallapatorius 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 ;-)

In math everything is deterministic, even the random variables! Mathematical function is just a mapping, it gives the same answer every time you plug in the same input. There are established methods of introducing uncertainty in this setting, however, the modeling tools are still deterministic.

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.

To give you yet another example, take an algorithm, a randomized one. But beware, the name may be misleading, there is no randomness there! What happens, is that, besides the normal data, the input of the algorithm consist also of random bits, that are commonly called "the seed". If the seed is fixed, the algorithm is not random at all! There may be some computation models that introduce randomness, but all this is just repackaging the old schema in the new wording (so is easier to work with).

Hope this helps ;-)

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Yes, that does help, but I'm trying to understand this in the context of (rudimentary) group theory. Does "deterministic" in that context simply mean that from any given state, any given series of basic actions such as "rotate, rotate, flip" will always lead to the same state—i.e. it means fully "determined" as opposed to "not determined completely" by the actions? If so, I can see that that follows directly from an operation always being a function (if that's what's going on). – raxacoricofallapatorius May 23 '12 at 21:41
@raxacoricofallapatorius Yeah, that's right. The rule of thumb: if you are not in the domain of probability theory (and you are not since you are dealing with group theory), unless stated otherwise, you can assume that everything is deterministic, that is, each operation is fully determined by the inputs. – dtldarek May 23 '12 at 21:51