  We identify compact groups of galaxies (CGs) within mock galaxy catalogues
from the Millennium Simulation at z=0 with three semi-analytic models (SAMs) of
galaxy formation. 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 the blending of close projected pairs. Half of the
mock CGs identified in projection contain at least 4 accordant velocities
(mvCGs), versus 70% for HCGs. In comparison to mvCGs, the HCGs are only 8%
complete at distances < 9000 km/s, missing the CGs with small angular sizes, a
strongly dominant galaxy, and (for one SAM) the CGs that are fainter and those
with lower surface brightness. We explore different ways to determine the
fraction of physically dense groups. 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 3D 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 smallest quartets within the CGs. We find that between 64% and
80% (depending on the SAM) of the mvCGs have 3D lengths shorter than 200 kpc/h,
between 71% and 80% have line-of-sight elongations less than 2, while between
59% and 76% have either 3D lengths shorter than 100 kpc/h or both lengths
shorter than 200 kpc/h and elongations smaller than 2. Therefore, chance
alignments (CAs) of galaxies concern at most 40% of the mvCGs. These CAs are
mostly produced from larger host groups, but a few have galaxies extending a
few Mpc beyond the host group. The mvCGs built with the Hickson selection with
(without) the close projected pair blending criterion have 10% higher (lower)
fractions of physically dense systems.
