  It is generally assumed that the central galaxy in a dark matter halo, that
is, the galaxy with the lowest specific potential energy, is also the brightest
halo galaxy (BHG), and that it resides at rest at the centre of the dark matter
potential well. This central galaxy paradigm (CGP) is an essential assumption
made in various fields of astronomical research. In this paper we test the
validity of the CGP using a large galaxy group catalogue constructed from the
Sloan Digital Sky Survey. For each group we compute two statistics, ${\cal R}$
and ${\cal S}$, which quantify the offsets of the line-of-sight velocities and
projected positions of brightest group galaxies relative to the other group
members. By comparing the cumulative distributions of $|{\cal R}|$ and $|{\cal
S}|$ to those obtained from detailed mock group catalogues, we rule out the
null-hypothesis that the CGP is correct. Rather, the data indicate that in a
non-zero fraction $f_{\rm BNC}(M)$ of all haloes of mass $M$ the BHG is not the
central galaxy, but instead, a satellite galaxy. In particular, we find that
$f_{\rm BNC}$ increases from $\sim 0.25$ in low mass haloes ($10^{12} h^{-1}
{\rm M_{\odot}} \leq M \lsim 2 \times 10^{13} h^{-1}{\rm M_{\odot}}$) to $\sim
0.4$ in massive haloes ($M \gsim 5 \times 10^{13} h^{-1} {\rm M_{\odot}}$). We
show that these values of $f_{\rm BNC}$ are uncomfortably high compared to
predictions from halo occupation statistics and from semi-analytical models of
galaxy formation. We end by discussing various implications of a non-zero
$f_{\rm BNC}(M)$, with an emphasis on the halo masses inferred from satellite
kinematics.
