Equivalence class relation for the associated bundle

Question

Let $P$ be a principal bundle with structure group $G$, $V$ a vector space and $\chi$ a group action \begin{equation} \chi:(P \times V) \times G \rightarrow P \times V, \quad(p, v, g) \mapsto(p, v) * g=\left(p * g, \rho\left(g^{-1}\right) v\right) \end{equation} Now one can define the equivalence relation \begin{equation} \left(p_{1}, v_{1}\right) \sim\left(p_{2}, v_{2}\right): \Leftrightarrow\left(p_{2}, v_{2}\right)=\left(p_{1}, v_{1}\right) * g \end{equation} with equivalence classes \begin{equation} [p, v]=\left\{\left(p * g, \rho\left(g^{-1}\right) v\right) \mid g \in G\right\} \end{equation} which will give rise to the associated bundle. Wikipedia (https://en.wikipedia.org/wiki/Associated_bundle) now claims that \begin{equation} [p * g, v]=[p, \rho(g) v] \end{equation} I probably miss something very trivial. But how does this relation follow form the definition above?

Answer 1

Write $\rho(h^{-1}) = \rho\left(h^{-1}g^{-1}g\right) = \rho\left((gh)^{-1}\right)\rho(g)$. It follows that: $$ \left( (p * g)*h, \rho(h^{-1})v \right) = \left(p*(gh), \rho\left((gh)^{-1}\right)\left(\rho(g)v\right) \right), $$ and the result follow from the fact $h \mapsto gh$ is a bijection of $G$.