Question: I was wondering about the ways we say something is a group. For example, the set $$G = \{z \in \mathbb{C} \mid z^n = 1 \text{ for some }n \in \mathbb{Z}^+\}$$ is a group under multiplication. But if I were to just say "Assume that $G = \{1,a,b,c\}$ is a group of order $4$," how can we not specify the binary operation? Is this just saying there exists a binary operation for which it is a group? 

**Edit:** Here is a definition of a group that seems to show what I am saying:

[![enter image description here](https://i.stack.imgur.com/oiB4T.png)](https://i.stack.imgur.com/oiB4T.png)

Answer: 
When you tell me "$G = \{1,a,b,c\}$", you have told me the operation is (if it is written at all, to be written as) multiplication (and that I should not assume the group is Abelian). You have told me this because you have written the identity element of the group as "$1$".

If instead you had told me "$G = \{0,a,b,c\}$", you would have told me that the group operation is (to be written as) addition (and that this group is likely Abelian). You have told me this because you have written the identity as "$0$".

Of course, one could write an abelian group multiplicatively and a nonAbelian group additively. One *can* do any number of things that are terrible ideas. Since I write so as not to confuse my reader, I wouldn't do either of those things.

I say "(to be written as)" in both cases, because there is nothing *intrinsic* about the operation that is used here. We could use any consistent symbol to write the group operation.  

$$ a \smile b, a \ast b, a \star b, a \oplus b, a \wr b, a \wedge b, \dots $$
In a group with operation written as multiplication, it is standard to not even write the operation; the group elements $\beta^{2}\alpha^{-1}$ and $\delta\beta\alpha$ recently showed up in my work. (These are written multiplicatively. The group is very non-Abelian (trivial center).)
"What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? "Oh, curse this maths" - concludes the student (who, possibly, becomes the Minister for Science in the future).

We get a totally different situation if we start off not with the group but with the concept of a transformation (a one-to-one mapping of a set onto itself) as it was historically. A collection of transformations of a set is called a group if along with any two transformations it contains the result of their consecutive application and an inverse transformation along with every transformation.

This is all the definition there is. The so-called "axioms" are in fact just (obvious) properties of groups of transformations."

From V. Arnol'd's article "On teaching mathematics", <http://pauli.uni-muenster.de/~munsteg/arnold.html>