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 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.