In the proof of Vitali's theorem the Vitali set is constructed. After the construction another set is created, namely $V_{k}=V+q_{k}=\{v+q_{k}:v\in V\}$ where $V$ ist the Vitali set and $q_k\in[0,1]$ are all the rational numbers in $[0,1]$.
Would it not be enough to just use the modulo group from which the Vitali set was created? Why is this additional step of creating $V_k$ necessary?