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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.

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

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