I am learning about representation theory, and I would like to be sure that I have properly understood the permutation representation. I therefore hope that someone can read what I have written below and tell me if I have misunderstood something.
Let $G$ be a group that acts on a set $S$ from the left, i.e., there exists a multiplication $G\times S\rightarrow S$ which is associative and such that $es=s$, with $s\in S$ and $e$ being the identity in $G$. We construct the permutation representation $\rho :G\rightarrow GL_n$ by defining a vector space $V$ by the span of the natural basis $\{\boldsymbol{e}_s\}_{s\in S}$ and then define each matrix $\rho(g)$, by $$\tag{1} \rho(g)\boldsymbol{e}_s:=\boldsymbol{e}_{gs}, $$ and extend linearly: $$\tag{2} \rho(g)\sum_{s\in S}a_s\boldsymbol{e}_s=\sum_{s\in S}a_s\rho(g)\boldsymbol{e}_s. $$ The associated module is defined by the vector space $V$ with the multiplication $g\boldsymbol{e}_s=\boldsymbol{e}_{gs}$. If $S=G$, then the permutation representation is also called the regular representation.