Q:

Why do we consider actions on vector spaces but not finite sets when doing representations of finite groups?

Say we have a finite group $G$, why we define a representation of it being a linear action on a finite dimensional vector space instead of on a finite set? As when we try to define a linear action on $V$, essentially we are defining that from one set of basis elements to another, and this job can be done by considering some actions on finite set $S$ as well. Thanks.

A:

Linear actions are quite special, and in some sense are "easier" to study than actions on sets, since there is a lot more structure on a vector space and we have many more tools for dealing with vector spaces. I once heard someone say that "eventually we would like to understand group actions on finite sets, but for now we'll settle for representation theory".
Not every linear action arises from an action on a finite set, because not every linear operator is simply a permutation of basis vectors. For example, consider the action of the cyclic group $\mathbb{Z} / n\mathbb{Z}$ on a finite set containing a single point: it must be the trivial action. However, there is an action of $\mathbb{Z} / n\mathbb{Z}$ on the one-dimensional vector space $\mathbb{C}$ by sending a generator to $\exp(2 \pi i / n)$.

