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We identify compact groups of galaxies (CGs) within mock galaxy catalogues from the Millennium Simulation at   z = 0  with the semi‐analytic models (SAMs) of galaxy formation of Bower et al., Croton et al. and De Lucia &amp; Blaizot. CGs are identified using the same 2D criteria as those visually applied by Hickson (1982) to his CGs (HCGs), but with a brightest galaxy magnitude limit, and we also add the important effect of observers blending close projected pairs. Half of the mock CGs identified in projection contain at least four accordant velocities ( m  v  CG s), versus 70 per cent for HCGs. In comparison to  m  v  CG s, the HCGs are only 8 per cent complete at distances  &lt;9000 km s −1  , missing the CGs with small angular sizes, a strongly dominant galaxy, and (for the second SAM) the  m  v  CG s that are fainter and those with lower surface brightness. 10 per cent of the mock  m  v  CG s are identical to the parent virialized group, meaning that they are isolated, while the remainder are embedded in their parent virialized groups. We explore different ways to determine the fraction of physically dense groups given the data from the simulations. Binding energy criteria turn out to be inapplicable given the segregation between galaxies and dark matter particles. We rely instead on the combination of the three‐dimensional length of the CGs (maximum real space galaxy separation) and their elongation along the line of sight (ratio of maximum line‐of‐sight to maximum projected separations), restricting ourselves in both cases to the smallest quartets within the CGs. We find that between 64 and 80 per cent (depending on the SAM) of the  m  v  CG s have 3D lengths shorter than  200  h  −1  kpc , between 71 and 80 per cent have line‐of‐sight elongations less than 2, while between 59 and 76 per cent have either 3D lengths shorter than  100  h  −1  kpc  or both lengths shorter than  200  h  −1  kpc  and elongations smaller than 2. Therefore, chance alignments (CAs) of galaxies concern at most 40 per cent of the  m  v  CG s. These CAs are mostly produced from larger host groups, but a few have galaxies extending a few Mpc beyond the host group. The  m  v  CG s built with the Hickson selection (respectively without the close projected pair blending criterion) have 10 per cent higher (lower) fractions of physically dense systems.

Compact groups from the Millennium Simulations – I. Their nature and the completeness of the Hickson sample