When we speak of a group, must we explicitly specify a certain binary operation? 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:
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.
"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