To be absolutely precise a group really is an order pair $(G, \cdot)$. (If you work in model theory, you may even want $(G,\cdot, e)$ to indicate the distinguished constant.)
The point is that you want to precisely define what is a group. But you notice that saying a group is just a set $G$, or the group is just a function $\cdot : G \times G \rightarrow G$ is not correct. Groups are not functions. In fact saying that a group is a set $G$ together with a function $\cdot : G \times G \rightarrow G$ is not even correct. (This is something called a structure in the language of groups.) Group even have to satisfy certain axioms. So a group really is a pair $(G, \cdot)$ which satisfies all the group axioms.
The above is a discussion about what a group really is. However, there is a distinction between the definition of a group and how much information you need so that most people would understand you in context. If you said "$\cdot : G \times G \rightarrow G$ is a group", I am sure most people would understand this to mean that $\cdot$ is the multiplication on a set $G$ which satisfy all the group properties.