Hi, I'm just doing a course in the Representation of Finite Groups, and in the notes, we are given the following definition of a representation which is not actually given a name:

Let G be a finite group, M a finite set, and let G act on M. For F some field we can construct a representation of G on

, FM having a basis

by

, extended to all of FM by linearity.

Several examples follow, e.g. where G is a group and M is a 1-element set, then FM is the trivial representation.

My notes are a little vague and I'm not exactly clear on the topic, so does anyone know what exactly this sort of representation is called so I can look at it in more depth online, or better yet have links to a source where I can research it?

Thanks