This may be a too simple question, but I am new to this representation theory stuff and just want to be clear.
In the Bruce E. Sagan "The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions", Definition 1.3.1 says that a vector space $V$ is a $G$-module (where $G$ is a given group) if there is a group homomorphism $\rho:G\rightarrow GL(V)$.
My question is: Does not that mean every vector space $V$ is automatically a $G$-module via the trivial homomorphism? Or should the definition be using the word "nontrivial"?
I might be missing something here and having a brain lag. I have read the previous section, but I am pretty sure it is presented differently in wording, defining that "A matrix representation of a group $G$ is a group homomorphism from $G$ to $GL_d$".
Thank you so much for the help.
The wording of the definition seems to imply that we only care whether the homomorphism exists, and if it does then we forget about the homomorphism and call the vector space a $G$-module. This does not tell the whole story. Even though every vector space is a $G$-module with the trivial homomorphism, this fact is not really of much importance. We actually care what the homomorphism specifically is, and the homomorphism should really be included in the definition of the module.