  Determining the positions of halo centres in large-scale structure surveys is
crucial for many cosmological studies. A common assumption is that halo centres
correspond to the location of their brightest member galaxies. In this paper,
we study the dynamics of brightest galaxies with respect to other halo members
in the Sloan Digital Sky Survey DR7. Specifically, we look at the line-of-sight
velocity and spatial offsets between brightest galaxies and their neighbours.
We compare those to detailed mock catalogues, constructed from high-resolution,
dark-matter-only $N$-body simulations, in which it is assumed that satellite
galaxies trace dark matter subhaloes. This allows us to place constraints on
the fraction $f_{\rm BNC}$ of haloes in which the brightest galaxy is not the
central. Compared to previous studies we explicitly take into account the
unrelaxed state of the host haloes, velocity offsets of halo cores and
correlations between $f_{\rm BNC}$ and the satellite occupation. We find that
$f_{\rm BNC}$ strongly decreases with the luminosity of the brightest galaxy
and increases with the mass of the host halo. Overall, in the halo mass range
$10^{13} - 10^{14.5} h^{-1} M_\odot$ we find $f_{\rm BNC} \sim 30\%$, in good
agreement with a previous study by Skibba et al. We discuss the implications of
these findings for studies inferring the galaxy--halo connection from satellite
kinematics, models of the conditional luminosity function and galaxy formation
in general.
