We trace the evolution of central galaxies in three ~10^13 M_sun galaxy groups simulated at high resolution in cosmological hydrodynamical simulations. The evolution in the group potential leads, at z=0, to central galaxies that are massive, gas-poor early-type systems supported by stellar velocity dispersion resembling either elliptical or S0 galaxies. Their z~2-2.5 main progenitors are massive M* ~ 3-10 x 10^10 M_sun, star forming (20-60 M_sun/yr) galaxies which host substantial reservoirs of cold gas (~5 x 10^9 M_sun) in extended gas disks. Our simulations thus show that star forming galaxies observed at z~2 are likely the main progenitors of central galaxies in galaxy groups at z=0. Their central stellar densities stay approximately constant from z~1.5 down to z=0. Instead, the galaxies grow inside-out, by acquiring a stellar envelope outside the innermost ~2 kpc. Consequently the density within the effective radius decreases by up to two orders of magnitude. Both major and minor mergers contribute to most of the mass accreted outside the effective radius and thus drive the evolution of the half-mass radii. In one of the three simulated groups the short central cooling time leads to a dramatic rejuvenation of the central group galaxy at z<1, affecting its morphology, kinematics and colors. This episode is eventually terminated by a group-group merger. Our simulations demonstrate that, in galaxy groups, the interplay between halo mass assembly, galaxy merging and gas accretion has a substantial influence on the star formation histories and z=0 morphologies of central galaxies.[Abridged]
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