The cold dark matter model has become the leading theoretical paradigm for the formation of structure in the Universe. Together with the theory of cosmic inflation, this model makes a clear prediction for the initial conditions for structure formation and predicts that structures grow hierarchically through gravitational instability. As a result, small structures collapse first and eventually build large structures such as groups and cluster of galaxies. While clusters are among the most massive bound structures in the Universe, groups are more numerous and most of the galaxies reside within galaxy groups. Testing this model requires that the precise measurements delivered by galaxy surveys can be compared to robust and equally precise theoretical models. The current project consists of two parts. In the first part, we investigate the existence and evolution of early-formed fossil galaxy groups, and the development of the luminosity gap between its brightest galaxies. We study the correlation of these properties with the group mass assembly history, by comparing observations to the Millennium simulation of dark matter particles, and the associated semi-analytic catalogues of galaxies, together with the Millennium gas simulation. Fossil Galaxy Groups are believed to be the end result of galaxies merging within a normal galaxy group, leaving behind the X-ray halo characteristic of a group. The sample of fossils in our study are selected according to the useful definition of fossil groups. The luminosity gap statistics in the Millennium Run are compared to the theoretical models. The study of the mass evolution of fossils shows that in comparison to normal groups, fossils are more evolved systems and have assembled their masses at higher redshifts, while normal groups are still evolving. Our work suggests the earlier formation and higher mass concentration of fossil systems. The estimated space densities from the Millennium Run are smaller (sometimes in agreement withi.