I posted a related question but was told to break it up into smaller pieces, so here goes.
I am reading Algebra by P. M. Cohn, and I am trying to wrap my head around modules and representations. The author wants to establish a link between the two concepts, and writes [...] take a finite-dimensional $G$-module $V$, with basis $v_1,...,v_d$ over field $k$. The action of $G$ on $V$ is completely described by the equations $$\tag{1} xv_i=\sum_j \rho_{i j}(x) v_j \quad(x \in G) $$ where $\rho_{ij}(x)\in k$, and it is easily checked that the matrices $\rho(x)=(\rho_{ij}(x))$ form a representation of $G$.
What I understand from the above is that since we assume that $V$ is a $G$-module, we know that there exists a multiplication $G\times V\rightarrow V$. Since the product $x v_i$ is in $V$, we know that we can write it as a linear combination of the basis vectors, which is what he does on the right-hand side of eq. $(1)$. I then want to show that the matrices so defined provide a representation. From what I understand, this is true if $$\tag{2} \forall x,y\in G :\rho(xy)=\rho(x)\rho(y)\quad \text{and}\quad \rho(\epsilon)=I, $$ where $\epsilon$ is the identity in $G$ and $I$ the identity matrix. From the definition of a $G$-module, we know that associativity and linearity must hold for the multiplication, so $$\tag{3} \begin{aligned} \sum_j \rho_{i j}(g h) v_j & =(g h) v_i,\quad \text{eq. (1)} \\ & =g\left(h v_i\right), \quad \text {associativity} \\ & =g \sum_j \rho_{i j}(h) v_j, \quad \text { eq. (1) } \qquad\qquad\qquad\qquad\qquad (3)\\ & =\sum_j \rho_{i j}(h) g v_j, \quad \text { linearity } \\ & =\sum_j \rho_{i j}(h) \sum_k \rho_{j k}(g) v_k \quad \text{eq.}\ (1) \end{aligned} $$ but this is where I get stuck. For eq. $(2)$ to hold, we should have $$\tag{4} \sum_j \rho_{i j}(g h) v_j=\sum_j \rho_{i j}(g)\rho_{i j}(h) v_j, $$ which does not seem to be the case considering eq. $(3)$. So what am I misunderstanding?
Equation (4) is wrong: the $ij$ entry of the product matrix $\rho(g)\rho(h)$ is not $\rho_{ij}(g)\rho_{ij}(h)$ (you don't multiply matrices by multiplying them entry-by-entry). Instead, the $ij$ entry of the product matrix is $\sum_k \rho_{ik}(g)\rho_{kj}(h)$, which exactly matches what you have in equation (3) (after a relabelling of the variables to swap $j$ and $k$ since there you are looking at the $ik$ entry rather than the $ij$ entry).