# A question about $G$-modules, representations, and $F[G]$-modules

I'll explain what I know, and then get to my question.

A group representation of a finite group $$G$$ is a group homomorphism $$\rho: G \to GL_F(V)$$ where $$V$$ is an $$F$$-vector space. Given a representation $$(\rho,V)$$ of $$G$$, we can define a linear action of $$G$$ on $$V$$ via $$gv := \rho(g)(v)$$. This gives the vector space $$V$$ the structure of a $$G$$-module.

Conversely, given a $$G$$-module structure on $$V$$, we can construct a group homomorphism $$\rho:G \to GL_F(V)$$ via $$g \mapsto \pi_g$$ where $$\pi_g(v) : = gv$$.

This establishes a bijection between $$G$$-modules over $$V$$ and group representations $$\rho:G \to GL_F(V)$$.

We can do a similar process to establish bijection between $$F[G]$$-modules and $$\rho:G \to GL_F(V)$$.

So we have essentially three viewpoints for a representation. My question is: why might one author choose to develop representation theory through $$F[G]$$-modules, while another chooses to develop the theory through $$G$$-modules? For example: Dummit and Foote chose the former, while Sagan chose the latter.

• Personal preference. The three approaches are completely equivalent. – egreg Aug 23 '20 at 21:20