Exploring the Equivalence of FG-Modules and Group Representations

[OMM!]

There is a Theorem that says FG-Modules are equivalent to group representations:


"(1) If $\rho$ is a representation of G over F and V = $F^{n}$, then V becomes an FG-Module if we define multiplication vg by: vg = v(g$\rho$), for all v in V, g in G.

(2) If V is an FG-Module and B a basis of V, then $\rho$: g $\mapsto$ $[g]_{B}$ is a representation of G over F, for all g in G"


I've been told and I have read that using FG-Modules is advantageous to using group representations, but what exactly is the advantage of this, other than getting results like Maschke's Theorem?!

Thanks for any help!

 

[fresh_42]

I don't see an advantage. It is simply another language, nothing else. And it is restricted to linear representations. We can use the language and theorems of ring theory, not only Maschke, which might be an advantage.