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We compare the mean mass assembly histories of compact and fossil galaxy groups in the Millennium dark matter simulation and an associated semi-analytic galaxy formation model. Tracing the halo mass of compact groups (CGs) from z=0 to z=1 shows that, on average, 55 per cent of the halo mass in compact groups is assembled since z~1, compared to 40 per cent of the halo mass in fossil groups (FGs) in the same time interval, indicating that compared to FGs, CGs are relatively younger galaxy systems. At z=0, for a given halo mass, fossil groups tend to have a larger concentration than compact groups. Investigating the evolution of CG's parameters show that they become more compact with time. CGs at z=0.5 see their magnitude gaps increase exponentially, but it takes ~10 Gyr for them to reach a magnitude gap of 2 magnitudes. The slow growth of the magnitude gap leads to only a minority (~41 per cent) of CGs selected at z=0.5 turning into a FG by z=0. Also, while three-quarters of FGs go through a compact phase, most fail to meet the CG isolation criterion, leaving only ~30 per cent of FGs fully satisfying the CG selection criteria. Therefore, there is no strong link of CGs turning into FGs or FGs originating from CGs. The relation between CGs and FGs is thus more complex, and in most cases, FGs and CGs follow different evolutionary tracks.
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