  It has been a long-standing question whether fossil groups are just sampling
the tail of the distribution of ordinary groups, or whether they are a
physically distinct class of objects, characterized by an unusual and special
formation history. To study this question, we here investigate fossil groups
identified in the hydrodynamical simulations of the GIMIC project, which
consists of resimulations of five regions in the Millennium Simulation (MS)
that are characterized by different large-scale densities, ranging from a deep
void to a proto-cluster region. For comparison, we also consider semi-analytic
models built on top of the MS, as well as a conditional luminosity function
approach. We identify galaxies in the GIMIC simulations as groups of stars and
use a spectral synthesis code to derive their optical properties. The X-ray
luminosity of the groups is estimated in terms of the thermal bremsstrahlung
emission of the gas in the host halos, neglecting metallicity effects. We focus
on comparing the properties of fossil groups in the theoretical models and
observational results, highlighting the differences between them, and trying to
identify possible dependencies on environment for which our approach is
particularly well set-up. We find that the optical fossil fraction in all of
our theoretical models declines with increasing halo mass, and there is no
clear environmental dependence. Combining the optical and X-ray selection
criteria for fossil groups, the halo mass dependence of the fossil groups seen
in optical vanishes. Over the GIMIC halo mass range we resolve best,
9.0\times1012 \sim 4.0\times1013 h-1 M, the central galaxies in the fossil
groups show similar properties as those in ordinary groups, in terms of age,
metallicity, color, concentration, and mass-to-light ratio. [abridged]
