Help showing associativity when multiplying group element by vector.

Question

Consider a representation $\rho=(\rho_{ij})$ of a group $G$ of degree $d$ and a $d$-dimensional vector space. I want to turn $V$ into a $G$-module by defining the action of $x\in G$ on each of the vectors of a basis of $V$ by $$ x \boldsymbol{v}_i=\sum_j \rho_{i j}(x) \boldsymbol{v}_j $$ and for a general $\boldsymbol{v}=\sum \alpha_i \boldsymbol{v}_i$ in $V$: $$ x\left(\sum \alpha_i \boldsymbol{v}_i\right)=\sum \alpha_i \rho_{i j}(x) \boldsymbol{v}_j $$ For $V$ to be a $G$-module I need the multiplication so defined to be associative, $(xy)\boldsymbol{v}=x(y\boldsymbol{v})$, however, I have a hard time showing. Consider $$ \begin{aligned} & (x y)\left(\sum_i \alpha_i \boldsymbol{v}_i\right)=\sum_{i, j} \alpha_i \rho_{i j}(x y) \boldsymbol{v}_j=\sum_{i, j, k} \alpha_i \rho_{i k}(x) \rho_{k j}(y) \boldsymbol{v}_j \\ & x\left(y \sum_i \alpha_i \boldsymbol{v}_i\right)=x \sum_{i, k} \alpha_i \rho_{i k}(y) \boldsymbol{v}_k=\sum_{i, k} \alpha_i \rho_{i k}(y) x \boldsymbol{v}_k=\sum_{i, j, k} \alpha_i \rho_{i k}(y) \rho_{k j}(x) \boldsymbol{v}_j \end{aligned} $$ These expressions don't seem to be identical, so what is wrong?

Answer 1

Replace $$x \boldsymbol{v}_i=\sum_j \rho_{i j}(x) \boldsymbol{v}_j$$ by $$x \boldsymbol{v}_j=\sum_i\rho_{i j}(x) \boldsymbol{v}_i.$$