Fossil groups (FGs) are galaxy systems with masses and X-ray haloes comparable to to those of groups and clusters of galaxies, but whose optical light is dominated by a single, large elliptical galaxy surrounded by much fainter companions. Their early assembly leaves enough time for L* galaxies to merge and makes FGs the oldest and most relaxed galaxy systems. Thus, FGs are the ideal environment to study the link between galaxy evolution and IGM. We propose to obtain deep X-ray observations in 4 FGs to measure their X-ray properties, and investigate: their scaling relations, the presence or absence of cool cores, the heating mechanisms and the metal enrichment history of the IGM. These observations will be compared with the status of the art of galaxy merging and cosmological simulations of FGs.