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: 
Yes. If you are assuming $G$ is a group, then it certainly must have the properties of a group by assumption. So, in particular, you are indeed assuming that $G$ must have such a binary operation, even if you do not explicitly state what it is.
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).)