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: 
In general, you are right. A group always has to be specified along with a binary operation, e.g. $(G,\times)$. But in many situations, the binary operations are understood. I'll give a few examples:

* $\mathbb{C}\backslash \{0\}$ would be a group with multiplication as its binary operation, because if it were addition, $0$ would be in the group.
* Group of 2x2 invertible matrices. In this case, we have either matrix multiplication or matrix addition as a choice for binary operation. But matrix addition would mean that the identity element is the zero matrix, which is not invertible.

Some others are just by convention, such as $\mathbb{Z}/n\mathbb{Z}$, the group of integer modulo n, which has the binary operation $+$. As compared to the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^\*$, which has the binary operation $\times$. 

That said, if there is any ambiguity at all, it is still necessary to state the binary operation.
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).)