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

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 $$x,y \in X$$, $$g \in G$$?

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 $$r$$.