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 an author talks of 4-element groups without specifying group operation, the statements made he is going to make expected to be valid for all groups of order 4 (there are just two actually). Then there is no need to go into specifics.

This not exactly about groups. When we make similar statements like "Let $X$ be topological space". The topology is not mentioned. So we do not know if it is a compact space or not. Do not know if connected or not. But the type of statement that author makes in that context is valid for all those cases. So that information is not needed.

Of course in your case there is one specific issue involved. Is there a group of order 4? Or else the whole discussion would be pointless. This, the author is expected to justify. But again in this case possibly the existence of finite groups of any given order was already established.
"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>