  We study the distribution and evolution of the stellar mass and the star
formation rate (SFR) of the brightest group galaxies (BGGs) over $ 0.04<z<1.3 $
using a large sample of $ 407 $ X-ray galaxy groups selected from the COSMOS,
AEGIS, and XMM-LSS fields. We compare our results with predictions from the
semi-analytic models based on the Millennium simulation. In contrast to model
predictions, we find that, as the universe evolves, the stellar mass
distribution evolves towards a normal distribution. This distribution tends to
skew to low mass BGGs at all redshifts implying the presence of a star-forming
population of the BGGs with $ M_S\sim10^{10.5} M_{\odot} $ which results in the
shape of the stellar mass distribution deviating from a normal distribution. In
agreement with models and previous studies, we find that the mean stellar mass
of BGGs grows with time by a factor of $\sim2$ between $z=1.3$ to $z=0.1$,
however, the significant growth occurs above $ z=0.4$. The BGGs are not
entirely a dormant population of galaxies, as low mass BGGs in low mass halos
are more active in forming stars than the BGGs in more massive halos, over the
same redshift range. We find that the average SFR of the BGGs evolves steeply
with redshift and fraction of the passive BGGs increases as a function of
increasing stellar mass and halo mass. Finally, we show that the specific SFR
of the BGGs within halos with $ M_{200} \leq 10^{13.4} M_{\odot} $ decreases
with increasing halo mass at $ z<0.4 $.
