We now introduce the concept of an FG-module, and show that there is a close connection between FG-modules and representations of G over F. Much of the material in the remainder of the book will be presented in terms of FG-modules, as there are several advantages to this approach to representation theory.
Let G be a group and let F be or .
Suppose that ρ: G → GL (n, F) is a representation of G. Write V = Fn, the vector space of all row vectors (λ1, . . . , λn) with λi ∈ F. For all v ∈ V and g ∈ G, the matrix product
of the ...