Are brightest halo galaxies central galaxies?

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, and , 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 and 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(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(BNC) increases from similar to 0.25 in low-mass haloes (1012 h-1 < M less than or similar to 2 x 1013 h-1 M-circle dot) to similar to 0.4 in massive haloes (M greater than or similar to 5 x 1013 h-1 M-circle dot). We show that these values of f(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(BNC)(M), with an emphasis on the halo masses inferred from satellite kinematics.