A question about "$\times$" notation with group actions

Question

Per Wikipedia, a (left) group action is defined as follows:

If $G $ is a group and $X$ is a set, then a (left) group action $φ$ of $G$ on $X$ is a function $$\varphi : G \times X \to X : (g,x)\mapsto \varphi(g,x)\cdots $$ (and so on).

My question is, what is the meaning of the "$\times$" notation in this context? It immediately calls to mind direct products and cross products. If so, it is defined via two groups, but $X$ here is a set than group, so I have my doubts.

Answer 1

It is the Cartesian product of (the underlying set of) $G$ with $X$, i.e., $$G\times X=\{(g, x)\mid g\in G, x\in X\}.$$

Answer 2

It is the Cartesian product of the two sets $G$ and $X$. The group structure on $G$ will then be used in the definition of an "action" (associativity and identity acts like an identity).

Answer 3

The $\times$ here is not a cross product, but a cartesian product (an operation between sets). Here it indicates that the action $\varphi$ is actually a function that has the set $G \times X$ for its domain ($G \times X$ is the set of all pairs of an element of $G$ together with an element of $X$). Its codomain is $X$.

Or put it more simple, $\phi$ takes one element of the group $G$ and one element of the set $X$, and returns as output an element of $X$.