  We present the first study of evolution of galaxy groups in the Illustris
simulation. We focus on dynamically relaxed and unrelaxed galaxy groups
representing dynamically evolved and evolving galaxy systems, respectively. The
evolutionary state of a group is probed from its luminosity gap and separation
between the brightest group galaxy and the center of mass of the group members.
We find that the Illustris simulation, over-produces large luminosity gap
galaxy systems, known as fossil systems, in comparison to observations and the
probed semi-analytical predictions. However, this simulation is equally
successful in recovering the correlation between luminosity gap and luminosity
centroid offset, in comparison to the probed semi-analytic model. We find
evolutionary tracks based on luminosity gap which indicate that a large
luminosity gap group is rooted in a small luminosity gap group, regardless of
the position of the brightest group galaxy within the halo. This simulation
helps, for the first time, to explore the black hole mass and its accretion
rate in galaxy groups. For a given stellar mass of the brightest group
galaxies, the black hole mass is larger in dynamically relaxed groups with a
lower rate of mass accretion. We find this consistent with the latest
observational studies of the radio activities in the brightest group galaxies
in fossil groups. We also find that the IGM in dynamically evolved groups is
hotter for a given halo mass than that in evolving groups, again consistent
with earlier observational studies.
