According to the current paradigm of galaxy formation, all galaxies form as a 
result of gas cooling at the center of the potential well of dark matter 
halos.  When a halo and its `central' galaxy is accreted by a larger halo, it 
becomes a subhalo and its galaxy becomes a `satellite' galaxy.  By analyzing 
the positions and velocities of brightest halo galaxies relative to the other 
galaxies occupying the halos, we test two related assumptions in this 
paradigm.  Firstly, the central galaxy in a halo is expected to be the most 
luminous and most massive galaxy, because it continues to accrete gas and 
grow while satellites merge into it.  Secondly, the brightest galaxy in a 
virialized halo is expected to reside at rest at the center of the potential 
well.  We use the Yang et al. (2007) galaxy group catalog extracted from the 
Sloan Digital Sky Survey, and we compare it to mock group catalogs.  Our 
analyses of the velocities and positions of group galaxies show that both of 
the assumptions are false.  In particular, we find that the dominant effect 
is that in a large fraction of groups and clusters the brightest galaxy is 
not the central galaxy.  We also quantify the degree to which central 
galaxies are offset and moving with respect to the halo center, although this 
is a weaker effect.