What does it mean for a group action to be linear?

Question

Let $G$ be a group and $X$ a set. The group $G$ acts on the set $X$ if there is a map $$G \times X \rightarrow X, (g,x) \rightarrow gx$$ such that $ex= x$, $h(gx) = (hg)(x)$ for all $x \in X$ and all $h,g \in G$, where $e$ is the identity of the group $G$.

Now, what does it mean for the action to be linear? What does it mean for the map $x \rightarrow gx$ to be linear? Does it mean $g(x + y) = gx + gy$ for all $x,y \in X$, $g \in G$?

Answer 1

The notion of linearity of a group action is only defined if $X$ is endowed with the additional structure of a vector space. So if the only structure on $X$ that is given to you is that of a bare set, as your question seems to imply, then linearity of the action is undefined.

But if $X$ is endowed with a vector space structure, then linearity of the action simply means that for each $g \in G$ the map $x \mapsto gx$ is a linear transformation of $X$. By definition this means $g(x+y) = gx+gy$ and $g(rx) = r g(x)$ for all $x,y \in X$ and all scalars $r$.