  The brightest galaxy in a dark matter halo is expected to reside at rest at
the center of the halo. In this paper we test this `Central Galaxy Paradigm'
using group catalogues extracted from the 2dFGRS and the SDSS. For each group
we compute a parameter, R, defined as the difference between the velocity of
the brightest group galaxy and the average velocity of the other group members
(hereafter satellites), normalized by the unbiased estimator of the velocity
dispersion of the satellite galaxies. Using detailed mock galaxy redshift
surveys, we show that the central galaxy paradigm is inconsistent with the data
at high confidence, and that instead the brightest halo galaxies have a
specific kinetic energy that is about 25 percent of that of the satellites.
This indicates that either central galaxies reside at the minimum of the dark
matter potential, but that the halo itself is not yet fully relaxed, or, that
the halo is relaxed, but that the central galaxy oscillates in its potential
well. The former is consistent with the fact that we find a weak hint that the
velocity bias of brightest halo galaxies is larger in more massive haloes,
while the latter may be indicative of cored, rather than cusped, dark matter
haloes. We discuss several implications of these findings, including mass
estimates based on satellite kinematics, strong gravitational lensing, halo
occupation models, and the frequency and longevity of lopsidedness in disk
galaxies.
