  We examine the statistical properties of the brightest group galaxies (BGGs)
using a complete spectroscopic sample of groups/clusters of galaxies selected
from the Data Release 7 of the Sloan Digital Sky Survey. We test whether BGGs
and other bright members of groups are consistent with an ordered population
among the total population of group galaxies. We find that the luminosity
distributions of BGGs do not follow the predictions from the order statistics
(OS). The average luminosities of BGGs are systematically brighter than OS
predictions. On the other hand, by properly taking into account the brightening
effect of the BGGs, the luminosity distributions of the second brightest
galaxies are in excellent agreement with the expectations of OS. The
brightening of BGGs relative to the OS expectation is consistent with a
scenario that the BGGs on average have over-grown about 20 percent masses
relative to the other member galaxies. The growth ($\Delta M$) is not
stochastic but correlated with the magnitude gap ($G_{1,2}$) between the
brightest and the second brightest galaxy. The growth ($\Delta M$) is larger
for the groups having more prominent BGGs (larger $G_{1,2}$) and averagely
contributes about 30 percent of the final $G_{1,2}$ of the groups of galaxies.
