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.
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