  Let $M_G$ denotes the centered Hardy-Littlewood maximal function associated
to the Carnot-Carath\'eodory distance or to the pseudo-distance associated to
the fundamental solution of the Grushin operator on $\R_x^n \times \R_u$,
$\Delta_G = \sum_{i = 1}^n \frac{\partial^2}{\partial x_i^2} + (\sum_{i = 1}^n
x_i^2) \frac{\partial^2}{\partial u^2}$. We get $L^p$ ($p > 1$) dimension free
estimates for $M_G$. We prove also that there exists a constant $A > 0$ such
that $\| M_G \|_{L^1 \longrightarrow L^{1, \infty}} \leq A n$, $\forall n$.
